kronecker product
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 296185 8 pageshttpdxdoiorg1011552013296185
Research ArticleOn the Kronecker Products and Their Applications
Huamin Zhang12 and Feng Ding1
1 Key Laboratory of Advanced Process Control for Light Industry of Ministry of Education Jiangnan UniversityWuxi 214122 China
2Department of Mathematics and Physics Bengbu College Bengbu 233030 China
Correspondence should be addressed to Feng Ding fdingjiangnaneducn
Received 10 March 2013 Accepted 6 June 2013
Academic Editor Song Cen
Copyright copy 2013 H Zhang and F Ding This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper studies the properties of the Kronecker product related to the mixed matrix products the vector operator and the vec-permutation matrix and gives several theorems and their proofs In addition we establish the relations between the singular valuesof two matrices and their Kronecker product and the relations between the determinant the trace the rank and the polynomialmatrix of the Kronecker products
1 Introduction
TheKronecker product named after GermanmathematicianLeopold Kronecker (December 7 1823ndashDecember 29 1891) isvery important in the areas of linear algebra and signal pro-cessing In fact the Kronecker product should be called theZehfuss product because Johann Georg Zehfuss published apaper in 1858 which contained the well-known determinantconclusion |A otimes B| = |A|
119899
|B|119898 for square matrices A and B
with order 119898 and 119899 [1]The Kronecker product has wide applications in system
theory [2ndash5] matrix calculus [6ndash9] matrix equations [10 11]system identification [12ndash15] and other special fields [16ndash19] Steeba and Wilhelm extended the exponential functionsformulas and the trace formulas of the exponential functionsof the Kronecker products [20] For estimating the upper andlower dimensions of the ranges of the two well-known lineartransformations T
1(X) = X minus AXB and T
2(X) = AX minus
XB Chuai and Tian established some rank equalities andinequalities for the Kronecker products [21] Correspondingto two different kinds ofmatrix partition KoningNeudeckerand Wansbeek developed two generalizations of the Kro-necker product and two related generalizations of the vectoroperator [22] The Kronecker product has an important rolein the linear matrix equation theory The solution of theSylvester and the Sylvester-like equations is a hotspot researcharea Recently the innovational and computationally efficient
numerical algorithms based on the hierarchical identificationprinciple for the generalized Sylvester matrix equations [23ndash25] and coupled matrix equations [10 26] were proposed byDing and Chen On the other hand the iterative algorithmsfor the extended Sylvester-conjugate matrix equations werediscussed in [27ndash29] Other related work is included in [30ndash32]
This paper establishes a new result about the singularvalue of the Kronecker product and gives a definition of thevec-permutation matrix In addition we prove the mixedproducts theorem and the conclusions on the vector operatorin a different method
This paper is organized as follows Section 2 gives the def-inition of the Kronecker product Section 3 lists some prop-erties based on the the mixed products theorem Section 4presents some interesting results about the vector operatorand the vec-permutation matrices Section 5 discusses thedeterminant trace and rank properties and the properties ofpolynomial matrices
2 The Definition and the Basic Properties ofthe Kronecker Product
Let F be a field such as R or C For any matrices A =
[119886119894119895] isin F119898times119899 and B isin F119901times119902 their Kronecker product
2 Journal of Applied Mathematics
(ie the direct product or tensor product) denoted as A otimesBis defined by
A otimes B = [119886119894119895B]
=
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
isin F(119898119901)times(119899119902)
(1)
It is clear that the Kronecker product of two diagonalmatrices is a diagonal matrix and the Kronecker product oftwo upper (lower) triangular matrices is an upper (lower)triangular matrix Let A119879 and A119867 denote the transpose andthe Hermitian transpose of matrix A respectively I
119898is
an identity matrix with order 119898 times 119898 The following basicproperties are obvious
Basic properties as follows
(1) I119898
otimes A = diag[AA A]
(2) if 120572 = [1198861 1198862 119886
119898]119879 and 120573 = [119887
1 1198872 119887
119899]119879 then
120572120573119879
= 120572 otimes 120573119879
= 120573119879
otimes 120572 isin F119898times119899
(3) if A = [A119894119895] is a block matrix then for any matrix B
A otimes B = [A119894119895otimes B]
(4) (120583A) otimes B = A otimes (120583B) = 120583(A otimes B)
(5) (A + B) otimes C = A otimes C + B otimes C
(6) A otimes (B + C) = A otimes B + A otimes C
(7) A otimes (B otimes C) = (A otimes B) otimes C = A otimes B otimes C
(8) (A otimes B)119879
= A119879 otimes B119879
(9) (A otimes B)119867
= A119867 otimes B119867
Property 2 indicates that 120572 and 120573119879 are commutativeProperty 7 shows that A otimes B otimes C is unambiguous
3 The Properties of the Mixed Products
This section discusses the properties based on the mixedproducts theorem [6 33 34]
Theorem 1 Let A isin F119898times119899 and B isin F119901times119902 then
A otimes B = (A otimes I119901) (I119899otimes B) = (I
119898otimes B) (A otimes I
119902) (2)
Proof According to the definition of the Kronecker productand the matrix multiplication we have
A otimes B =
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
B 0 sdot sdot sdot 00 B sdot sdot sdot 0
d
0 0 sdot sdot sdot B
]]]]
]
= (A otimes I119901) (I119899otimes B)
A otimes B =
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
=
[[[[
[
B 0 sdot sdot sdot 00 B sdot sdot sdot 0
d
0 0 sdot sdot sdot B
]]]]
]
[[[[
[
11988611I119902
11988612I119902
sdot sdot sdot 1198861119899I119902
11988621I119902
11988622I119902
sdot sdot sdot 1198862119899I119902
1198861198981I119902
1198861198982I119902
sdot sdot sdot 119886119898119899I119902
]]]]
]
= (I119898
otimes B) (A otimes I119902)
(3)
FromTheorem 1 we have the following corollary
Corollary 2 Let A isin F119898times119898 and B isin F119899times119899 Then
A otimes B = (A otimes I119899) (I119898
otimes B) = (I119898
otimes B) (A otimes I119899) (4)
This mean that I119898
otimes B and A otimes I119899are commutative for square
matrices A and B
Using Theorem 1 we can prove the following mixedproducts theorem
Theorem 3 Let A = [119886119894119895] isin F119898times119899 C = [119888
119894119895] isin F119899times119901 B isin F119902times119903
and D isin F 119903times119904 Then
(A otimes B) (C otimes D) = (AC) otimes (BD) (5)
Proof According toTheorem 1 we have
(A otimes B) (C otimes D)
= (A otimes I119902) (I119899otimes B) (C otimes I
119903) (I119901otimes D)
= (A otimes I119902) [(I119899otimes B) (C otimes I
119903)] (I119901otimes D)
= (A otimes I119902) (C otimes B) (I
119901otimes D)
= (A otimes I119902) [(C otimes I
119902) (I119901otimes B)] (I
119901otimes D)
Journal of Applied Mathematics 3
= [(A otimes I119902) (C otimes I
119902)] [(I
119901otimes B) (I
119901otimes D)]
= [(AC) otimes I119902)] [I119901otimes (BD)]
= (AC) otimes (BD) (6)
Let A[1] = A and define the Kronecker power by
A[119896+1] = A[119896] otimes A = A otimes A[119896] 119896 = 1 2 (7)
FromTheorem 3 we have the following corollary [7]
Corollary 4 If the following matrix products exist then onehas
(1) (A1otimes B1)(A2otimes B2) sdot sdot sdot (A
119901otimes B119901) = (A
1A2sdot sdot sdotA119901) otimes
(B1B2sdot sdot sdotB119901)
(2) (A1otimes A2otimes sdot sdot sdot otimes A
119901)(B1otimes B2otimes sdot sdot sdot otimes B
119901) = (A
1B1) otimes
(A2B2) otimes sdot sdot sdot otimes (A
119901B119901)
(3) [AB][119896]
= A[119896]B[119896]
A squarematrixA is said to be a normalmatrix if and onlyif A119867A = AA119867 A square matrix A is said to be a unitarymatrix if and only if A119867A = AA119867 = I Straightforwardcalculation gives the following conclusions [6 7 33 34]
Theorem 5 For any square matrices A and B(1) if Aminus1 and Bminus1 exist then (A otimes B)
minus1
= Aminus1 otimes Bminus1(2) ifA and B are normal matrices thenAotimesB is a normal
matrix(3) if A and B are unitary (orthogonal) matrices then A otimes
B is a unitary (orthogonal) matrix
Let 120582[A] = 1205821 1205822 120582
119898 denote the eigenvalues of A
and let 120590[A] = 1205901 1205902 120590
119903 denote the nonzero singular
values ofA According to the definition of the eigenvalue andTheorem 3 we have the following conclusions [34]
Theorem 6 Let A isin F119898times119898 and B isin F119899times119899 119896 and 119897 are positiveintegers Then 120582[A119896 otimes B119897] = 120582
119896
119894120583119897
119895| 119894 = 1 2 119898 119895 =
1 2 119899 = 120582[B119897 otimes A119896] Here 120582[A] = 1205821 1205822 120582
119898 and
120582[B] = 1205831 1205832 120583
119899
According to the definition of the singular value andTheorem 3 for any matrices A and B we have the nexttheorem
Theorem 7 Let A isin F119898times119899 and B isin F119901times119902 If rank[A] = 119903120590[A] = 120590
1 1205902 120590
119903 rank[B] = 119904 and 120590[B] =
1205881 1205882 120588
119904 then 120590[A otimes B] = 120590
119894120588119895
| 119894 = 1 2 119903 119895 =
1 2 119904 = 120590[B otimes A]
Proof According to the singular value decomposition theo-rem there exist the unitary matrices U V and W Q whichsatisfy
A = U [Σ 00 0]V B = [
Γ 00 0]Q (8)
where Σ = diag[1205901 1205902 120590
119903] and Γ = diag[120588
1 1205882 120588
119904]
According to Corollary 4 we have
A otimes B = U [Σ 00 0]V otimes W [
Γ 00 0]Q
= (U otimes W) [Σ 00 0] otimes [
Γ 00 0] (V otimes Q)
= (U otimes W) [
[
Σ otimes [Σ 00 0] 00 0
]
]
(V otimes Q)
= (U otimes W) [Σ otimes Γ 00 0] (V otimes Q)
(9)
Since U otimes W and V otimes Q are unitary matrices and Σ otimes Γ =
diag[12059011205881 12059011205882 120590
1120588119904 120590
119903120588119904] this proves the theorem
According toTheorem 7 we have the next corollary
Corollary 8 For any matricesA B andC one has 120590[AotimesBotimes
C] = 120590[C otimes B otimes A]
4 The Properties of the Vector Operator andthe Vec-Permutation Matrix
In this section we introduce a vector-valued operator and avec-permutation matrix
Let A = [a1 a2 a
119899] isin F119898times119899 where a
119895isin F119898 119895 =
1 2 119899 then the vector col[A] is defined by
col [A] =
[[[[
[
a1
a2
a119899
]]]]
]
isin F119898119899
(10)
Theorem 9 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119899 Then
(1) (I119901otimes A)col[B] = col[AB]
(2) (A otimes I119901)col[C] = col[CAT
]
Proof Let (B)119894denote the 119894th column of matrix B we have
(I119901otimes A) col [B] =
[[[[
[
A 0 sdot sdot sdot 00 A sdot sdot sdot 0
d
0 0 sdot sdot sdot A
]]]]
]
[[[[
[
(B)1
(B)2
(B)119901
]]]]
]
=
[[[[
[
A(B)1
A(B)2
A(B)119901
]]]]
]
=
[[[[
[
(AB)1
(AB)2
(AB)119901
]]]]
]
= col [AB]
(11)
4 Journal of Applied Mathematics
Similarly we have
(A otimes I119901) col [C]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
(C)1
(C)2
(C)119899
]]]]
]
=
[[[[
[
11988611(C)1+ 11988612(C)2+ sdot sdot sdot + 119886
1119899(C)119899
11988621(C)1+ 11988622(C)2+ sdot sdot sdot + 119886
2119899(C)119899
1198861198981
(C)1+ 1198861198982
(C)2+ sdot sdot sdot + 119886
119898119899(C)119899
]]]]
]
=
[[[[[
[
C(A119879)1
C(A119879)2
C(A119879)
119898
]]]]]
]
=
[[[[[
[
(CA119879)1
(CA119879)2
(CA119879)
119898
]]]]]
]
= col [CA119879]
(12)
Theorem 10 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119902 Then
col [ABC] = (C119879 otimes A) col [B] (13)
Proof According toTheorems 9 and 1 we have
col [ABC] = col [(AB)C]
= (C119879 otimes I119898) col [AB]
= (C119879 otimes I119898) (I119901otimes A) col [B]
= [(C119879 otimes I119898) (I119901otimes A)] col [B]
= (C119879 otimes A) col [B]
(14)
Theorem 10 plays an important role in solving the matrixequations [25 35ndash37] system identification [38ndash54] andcontrol theory [55ndash58]
Let e119894119899denote an 119899-dimensional column vector which has
1 in the 119894th position and 0rsquos elsewhere that is
e119894119899
= [0 0 0 1 0 0]119879
(15)
Define the vec-permutation matrix
P119898119899
=
[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]
]
isin R119898119899times119898119899
(16)
which can be expressed as [6 7 33 37]119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
(17)
These two definitions of the vec-permutation matrix areequivalent that is
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
= P119898119899
(18)
In fact according to Theorem 3 and the basic properties ofthe Kronecker product we have
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119879119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899e119879119895119898
) otimes (e119895119898e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119879119895119898
) otimes (e119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
[e119896119899
otimes (e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
=
119899
sum119896=1
[
[
e119896119899
otimes
119898
sum119895=1
(e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
]
=
119899
sum119896=1
[e119896119899
otimes I119898
otimes e119879119896119899
]
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= P119898119899
(19)
Based on the definition of the vec-permutationmatrix wehave the following conclusions
Theorem 11 According to the definition of P119898119899 one has
(1) P119879119898119899
= P119899119898
(2) P119879119898119899P119898119899
= P119898119899P119879119898119899
= I119898119899
That is P119898119899
is an (119898119899) times (119898119899) permutation matrix
Journal of Applied Mathematics 5
Proof According to the definition ofP119898119899Theorem 3 and the
basic properties of the Kronecker product we have
P119879119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
119879
= [I119879119898
otimes e1119899 I119879119898
otimes e2119899 I119879
119898otimes e119899119899
]
=
[[[[[[
[
e1198791119898
otimes e1119899
e1198791119898
otimes e2119899
sdot sdot sdot e1198791119898
otimes e119899119899
e1198792119898
otimes e1119899
e1198792119898
otimes e2119899
sdot sdot sdot e1198792119898
otimes e119899119899
e119879119898119898
otimes e1119899
e119879119898119898
otimes e2119899
sdot sdot sdot e119879119898119898
otimes e119899119899
]]]]]]
]
=
[[[[[[
[
e1119899
otimes e1198791119898
e2119899
otimes e1198791119898
sdot sdot sdot e119899119899
otimes e1198791119898
e1119899
otimes e1198792119898
e2119899
otimes e1198792119898
sdot sdot sdot e119899119899
otimes e1198792119898
e1119899
otimes e119879119898119898
e2119899
otimes e119879119898119898
sdot sdot sdot e119899119899
otimes e119879119898119898
]]]]]]
]
=
[[[[[[
[
I119899otimes e1198791119898
I119899otimes e1198792119898
I119899otimes e119879119898119898
]]]]]]
]
= P119899119898
(20)
P119898119899P119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
[I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
=
[[[[[[[
[
I119898
otimes (e1198791119899e1119899) I119898
otimes (e1198791119899e2119899) sdot sdot sdot I
119898otimes (e1198791119899e119899119899
)
I119898
otimes (e1198792119899e1119899) I119898
otimes (e1198792119899e2119899) sdot sdot sdot I
119898otimes (e1198792119899e119899119899
)
I119898
otimes (e119879119899119899e1119899) I119898
otimes (e119879119899119899e2119899) sdot sdot sdot I
119898otimes (e119879119899119899e119899119899
)
]]]]]]]
]
=
[[[[
[
I119898
0 sdot sdot sdot 00 I119898
sdot sdot sdot 0
d
0 0 sdot sdot sdot I119898
]]]]
]
= I119898119899
(21)
P119879119898119899P119898119899
= [I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= I119898
otimes [
119899
sum119894=1
e119894119899e119879119894119899]
= I119898
otimes I119899
= I119898119899
(22)
For any matrix A isin F119898times119899 we have col [A] = P119898119899col [A119879]
Theorem 12 IfA isin F119898times119899 andB isin F119901times119902 then one hasP119898119901
(Aotimes
B)P119879119899119902
= B otimes A
Proof Let B = [119887119894119895] = [
[
B1B2B119901
]
]
where B119894 isin F1times119902 and 119894 =
1 2 119901 and 119895 = 1 2 119902 According to the definition ofP119898119899
and the Kronecker product we have
P119898119901
(A otimes B)P119879119899119902
=
[[[[[[[
[
I119898
otimes e1198791119901
I119898
otimes e1198792119901
I119898
otimes e119879119901119901
]]]]]]]
]
[(A)1otimes B (A)
2otimes B (A)
119899otimes B]P119879
119899119902
=
[[[[
[
(A)1otimes B1 (A)
2otimes B1 sdot sdot sdot (A)
119899otimes B1
(A)1otimes B2 (A)
2otimes B2 sdot sdot sdot (A)
119899otimes B2
(A)1otimes B119901 (A)
2otimes B119901 sdot sdot sdot (A)
119899otimes B119901
]]]]
]
P119879119899119902
=
[[[[
[
A otimes B1A otimes B2
A otimes B119901
]]]]
]
[I119899otimes e1119902 I119899otimes e2119902 I
119899otimes e119902119902]
=
[[[[
[
A11988711
A11988712
sdot sdot sdot A1198871119902
A11988721
A11988722
sdot sdot sdot A1198872119902
A1198871199011
A1198871199012
sdot sdot sdot A119887119901119902
]]]]
]
= B otimes A
(23)
FromTheorem 12 we have the following corollaries
Corollary 13 If A isin F119898times119899 then P119898119903
(A otimes I119903)P119879119899119903
= I119903otimes A
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
(ie the direct product or tensor product) denoted as A otimesBis defined by
A otimes B = [119886119894119895B]
=
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
isin F(119898119901)times(119899119902)
(1)
It is clear that the Kronecker product of two diagonalmatrices is a diagonal matrix and the Kronecker product oftwo upper (lower) triangular matrices is an upper (lower)triangular matrix Let A119879 and A119867 denote the transpose andthe Hermitian transpose of matrix A respectively I
119898is
an identity matrix with order 119898 times 119898 The following basicproperties are obvious
Basic properties as follows
(1) I119898
otimes A = diag[AA A]
(2) if 120572 = [1198861 1198862 119886
119898]119879 and 120573 = [119887
1 1198872 119887
119899]119879 then
120572120573119879
= 120572 otimes 120573119879
= 120573119879
otimes 120572 isin F119898times119899
(3) if A = [A119894119895] is a block matrix then for any matrix B
A otimes B = [A119894119895otimes B]
(4) (120583A) otimes B = A otimes (120583B) = 120583(A otimes B)
(5) (A + B) otimes C = A otimes C + B otimes C
(6) A otimes (B + C) = A otimes B + A otimes C
(7) A otimes (B otimes C) = (A otimes B) otimes C = A otimes B otimes C
(8) (A otimes B)119879
= A119879 otimes B119879
(9) (A otimes B)119867
= A119867 otimes B119867
Property 2 indicates that 120572 and 120573119879 are commutativeProperty 7 shows that A otimes B otimes C is unambiguous
3 The Properties of the Mixed Products
This section discusses the properties based on the mixedproducts theorem [6 33 34]
Theorem 1 Let A isin F119898times119899 and B isin F119901times119902 then
A otimes B = (A otimes I119901) (I119899otimes B) = (I
119898otimes B) (A otimes I
119902) (2)
Proof According to the definition of the Kronecker productand the matrix multiplication we have
A otimes B =
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
B 0 sdot sdot sdot 00 B sdot sdot sdot 0
d
0 0 sdot sdot sdot B
]]]]
]
= (A otimes I119901) (I119899otimes B)
A otimes B =
[[[[
[
11988611B 11988612B sdot sdot sdot 119886
1119899B
11988621B 11988622B sdot sdot sdot 119886
2119899B
1198861198981B 1198861198982B sdot sdot sdot 119886
119898119899B
]]]]
]
=
[[[[
[
B 0 sdot sdot sdot 00 B sdot sdot sdot 0
d
0 0 sdot sdot sdot B
]]]]
]
[[[[
[
11988611I119902
11988612I119902
sdot sdot sdot 1198861119899I119902
11988621I119902
11988622I119902
sdot sdot sdot 1198862119899I119902
1198861198981I119902
1198861198982I119902
sdot sdot sdot 119886119898119899I119902
]]]]
]
= (I119898
otimes B) (A otimes I119902)
(3)
FromTheorem 1 we have the following corollary
Corollary 2 Let A isin F119898times119898 and B isin F119899times119899 Then
A otimes B = (A otimes I119899) (I119898
otimes B) = (I119898
otimes B) (A otimes I119899) (4)
This mean that I119898
otimes B and A otimes I119899are commutative for square
matrices A and B
Using Theorem 1 we can prove the following mixedproducts theorem
Theorem 3 Let A = [119886119894119895] isin F119898times119899 C = [119888
119894119895] isin F119899times119901 B isin F119902times119903
and D isin F 119903times119904 Then
(A otimes B) (C otimes D) = (AC) otimes (BD) (5)
Proof According toTheorem 1 we have
(A otimes B) (C otimes D)
= (A otimes I119902) (I119899otimes B) (C otimes I
119903) (I119901otimes D)
= (A otimes I119902) [(I119899otimes B) (C otimes I
119903)] (I119901otimes D)
= (A otimes I119902) (C otimes B) (I
119901otimes D)
= (A otimes I119902) [(C otimes I
119902) (I119901otimes B)] (I
119901otimes D)
Journal of Applied Mathematics 3
= [(A otimes I119902) (C otimes I
119902)] [(I
119901otimes B) (I
119901otimes D)]
= [(AC) otimes I119902)] [I119901otimes (BD)]
= (AC) otimes (BD) (6)
Let A[1] = A and define the Kronecker power by
A[119896+1] = A[119896] otimes A = A otimes A[119896] 119896 = 1 2 (7)
FromTheorem 3 we have the following corollary [7]
Corollary 4 If the following matrix products exist then onehas
(1) (A1otimes B1)(A2otimes B2) sdot sdot sdot (A
119901otimes B119901) = (A
1A2sdot sdot sdotA119901) otimes
(B1B2sdot sdot sdotB119901)
(2) (A1otimes A2otimes sdot sdot sdot otimes A
119901)(B1otimes B2otimes sdot sdot sdot otimes B
119901) = (A
1B1) otimes
(A2B2) otimes sdot sdot sdot otimes (A
119901B119901)
(3) [AB][119896]
= A[119896]B[119896]
A squarematrixA is said to be a normalmatrix if and onlyif A119867A = AA119867 A square matrix A is said to be a unitarymatrix if and only if A119867A = AA119867 = I Straightforwardcalculation gives the following conclusions [6 7 33 34]
Theorem 5 For any square matrices A and B(1) if Aminus1 and Bminus1 exist then (A otimes B)
minus1
= Aminus1 otimes Bminus1(2) ifA and B are normal matrices thenAotimesB is a normal
matrix(3) if A and B are unitary (orthogonal) matrices then A otimes
B is a unitary (orthogonal) matrix
Let 120582[A] = 1205821 1205822 120582
119898 denote the eigenvalues of A
and let 120590[A] = 1205901 1205902 120590
119903 denote the nonzero singular
values ofA According to the definition of the eigenvalue andTheorem 3 we have the following conclusions [34]
Theorem 6 Let A isin F119898times119898 and B isin F119899times119899 119896 and 119897 are positiveintegers Then 120582[A119896 otimes B119897] = 120582
119896
119894120583119897
119895| 119894 = 1 2 119898 119895 =
1 2 119899 = 120582[B119897 otimes A119896] Here 120582[A] = 1205821 1205822 120582
119898 and
120582[B] = 1205831 1205832 120583
119899
According to the definition of the singular value andTheorem 3 for any matrices A and B we have the nexttheorem
Theorem 7 Let A isin F119898times119899 and B isin F119901times119902 If rank[A] = 119903120590[A] = 120590
1 1205902 120590
119903 rank[B] = 119904 and 120590[B] =
1205881 1205882 120588
119904 then 120590[A otimes B] = 120590
119894120588119895
| 119894 = 1 2 119903 119895 =
1 2 119904 = 120590[B otimes A]
Proof According to the singular value decomposition theo-rem there exist the unitary matrices U V and W Q whichsatisfy
A = U [Σ 00 0]V B = [
Γ 00 0]Q (8)
where Σ = diag[1205901 1205902 120590
119903] and Γ = diag[120588
1 1205882 120588
119904]
According to Corollary 4 we have
A otimes B = U [Σ 00 0]V otimes W [
Γ 00 0]Q
= (U otimes W) [Σ 00 0] otimes [
Γ 00 0] (V otimes Q)
= (U otimes W) [
[
Σ otimes [Σ 00 0] 00 0
]
]
(V otimes Q)
= (U otimes W) [Σ otimes Γ 00 0] (V otimes Q)
(9)
Since U otimes W and V otimes Q are unitary matrices and Σ otimes Γ =
diag[12059011205881 12059011205882 120590
1120588119904 120590
119903120588119904] this proves the theorem
According toTheorem 7 we have the next corollary
Corollary 8 For any matricesA B andC one has 120590[AotimesBotimes
C] = 120590[C otimes B otimes A]
4 The Properties of the Vector Operator andthe Vec-Permutation Matrix
In this section we introduce a vector-valued operator and avec-permutation matrix
Let A = [a1 a2 a
119899] isin F119898times119899 where a
119895isin F119898 119895 =
1 2 119899 then the vector col[A] is defined by
col [A] =
[[[[
[
a1
a2
a119899
]]]]
]
isin F119898119899
(10)
Theorem 9 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119899 Then
(1) (I119901otimes A)col[B] = col[AB]
(2) (A otimes I119901)col[C] = col[CAT
]
Proof Let (B)119894denote the 119894th column of matrix B we have
(I119901otimes A) col [B] =
[[[[
[
A 0 sdot sdot sdot 00 A sdot sdot sdot 0
d
0 0 sdot sdot sdot A
]]]]
]
[[[[
[
(B)1
(B)2
(B)119901
]]]]
]
=
[[[[
[
A(B)1
A(B)2
A(B)119901
]]]]
]
=
[[[[
[
(AB)1
(AB)2
(AB)119901
]]]]
]
= col [AB]
(11)
4 Journal of Applied Mathematics
Similarly we have
(A otimes I119901) col [C]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
(C)1
(C)2
(C)119899
]]]]
]
=
[[[[
[
11988611(C)1+ 11988612(C)2+ sdot sdot sdot + 119886
1119899(C)119899
11988621(C)1+ 11988622(C)2+ sdot sdot sdot + 119886
2119899(C)119899
1198861198981
(C)1+ 1198861198982
(C)2+ sdot sdot sdot + 119886
119898119899(C)119899
]]]]
]
=
[[[[[
[
C(A119879)1
C(A119879)2
C(A119879)
119898
]]]]]
]
=
[[[[[
[
(CA119879)1
(CA119879)2
(CA119879)
119898
]]]]]
]
= col [CA119879]
(12)
Theorem 10 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119902 Then
col [ABC] = (C119879 otimes A) col [B] (13)
Proof According toTheorems 9 and 1 we have
col [ABC] = col [(AB)C]
= (C119879 otimes I119898) col [AB]
= (C119879 otimes I119898) (I119901otimes A) col [B]
= [(C119879 otimes I119898) (I119901otimes A)] col [B]
= (C119879 otimes A) col [B]
(14)
Theorem 10 plays an important role in solving the matrixequations [25 35ndash37] system identification [38ndash54] andcontrol theory [55ndash58]
Let e119894119899denote an 119899-dimensional column vector which has
1 in the 119894th position and 0rsquos elsewhere that is
e119894119899
= [0 0 0 1 0 0]119879
(15)
Define the vec-permutation matrix
P119898119899
=
[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]
]
isin R119898119899times119898119899
(16)
which can be expressed as [6 7 33 37]119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
(17)
These two definitions of the vec-permutation matrix areequivalent that is
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
= P119898119899
(18)
In fact according to Theorem 3 and the basic properties ofthe Kronecker product we have
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119879119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899e119879119895119898
) otimes (e119895119898e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119879119895119898
) otimes (e119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
[e119896119899
otimes (e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
=
119899
sum119896=1
[
[
e119896119899
otimes
119898
sum119895=1
(e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
]
=
119899
sum119896=1
[e119896119899
otimes I119898
otimes e119879119896119899
]
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= P119898119899
(19)
Based on the definition of the vec-permutationmatrix wehave the following conclusions
Theorem 11 According to the definition of P119898119899 one has
(1) P119879119898119899
= P119899119898
(2) P119879119898119899P119898119899
= P119898119899P119879119898119899
= I119898119899
That is P119898119899
is an (119898119899) times (119898119899) permutation matrix
Journal of Applied Mathematics 5
Proof According to the definition ofP119898119899Theorem 3 and the
basic properties of the Kronecker product we have
P119879119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
119879
= [I119879119898
otimes e1119899 I119879119898
otimes e2119899 I119879
119898otimes e119899119899
]
=
[[[[[[
[
e1198791119898
otimes e1119899
e1198791119898
otimes e2119899
sdot sdot sdot e1198791119898
otimes e119899119899
e1198792119898
otimes e1119899
e1198792119898
otimes e2119899
sdot sdot sdot e1198792119898
otimes e119899119899
e119879119898119898
otimes e1119899
e119879119898119898
otimes e2119899
sdot sdot sdot e119879119898119898
otimes e119899119899
]]]]]]
]
=
[[[[[[
[
e1119899
otimes e1198791119898
e2119899
otimes e1198791119898
sdot sdot sdot e119899119899
otimes e1198791119898
e1119899
otimes e1198792119898
e2119899
otimes e1198792119898
sdot sdot sdot e119899119899
otimes e1198792119898
e1119899
otimes e119879119898119898
e2119899
otimes e119879119898119898
sdot sdot sdot e119899119899
otimes e119879119898119898
]]]]]]
]
=
[[[[[[
[
I119899otimes e1198791119898
I119899otimes e1198792119898
I119899otimes e119879119898119898
]]]]]]
]
= P119899119898
(20)
P119898119899P119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
[I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
=
[[[[[[[
[
I119898
otimes (e1198791119899e1119899) I119898
otimes (e1198791119899e2119899) sdot sdot sdot I
119898otimes (e1198791119899e119899119899
)
I119898
otimes (e1198792119899e1119899) I119898
otimes (e1198792119899e2119899) sdot sdot sdot I
119898otimes (e1198792119899e119899119899
)
I119898
otimes (e119879119899119899e1119899) I119898
otimes (e119879119899119899e2119899) sdot sdot sdot I
119898otimes (e119879119899119899e119899119899
)
]]]]]]]
]
=
[[[[
[
I119898
0 sdot sdot sdot 00 I119898
sdot sdot sdot 0
d
0 0 sdot sdot sdot I119898
]]]]
]
= I119898119899
(21)
P119879119898119899P119898119899
= [I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= I119898
otimes [
119899
sum119894=1
e119894119899e119879119894119899]
= I119898
otimes I119899
= I119898119899
(22)
For any matrix A isin F119898times119899 we have col [A] = P119898119899col [A119879]
Theorem 12 IfA isin F119898times119899 andB isin F119901times119902 then one hasP119898119901
(Aotimes
B)P119879119899119902
= B otimes A
Proof Let B = [119887119894119895] = [
[
B1B2B119901
]
]
where B119894 isin F1times119902 and 119894 =
1 2 119901 and 119895 = 1 2 119902 According to the definition ofP119898119899
and the Kronecker product we have
P119898119901
(A otimes B)P119879119899119902
=
[[[[[[[
[
I119898
otimes e1198791119901
I119898
otimes e1198792119901
I119898
otimes e119879119901119901
]]]]]]]
]
[(A)1otimes B (A)
2otimes B (A)
119899otimes B]P119879
119899119902
=
[[[[
[
(A)1otimes B1 (A)
2otimes B1 sdot sdot sdot (A)
119899otimes B1
(A)1otimes B2 (A)
2otimes B2 sdot sdot sdot (A)
119899otimes B2
(A)1otimes B119901 (A)
2otimes B119901 sdot sdot sdot (A)
119899otimes B119901
]]]]
]
P119879119899119902
=
[[[[
[
A otimes B1A otimes B2
A otimes B119901
]]]]
]
[I119899otimes e1119902 I119899otimes e2119902 I
119899otimes e119902119902]
=
[[[[
[
A11988711
A11988712
sdot sdot sdot A1198871119902
A11988721
A11988722
sdot sdot sdot A1198872119902
A1198871199011
A1198871199012
sdot sdot sdot A119887119901119902
]]]]
]
= B otimes A
(23)
FromTheorem 12 we have the following corollaries
Corollary 13 If A isin F119898times119899 then P119898119903
(A otimes I119903)P119879119899119903
= I119903otimes A
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
= [(A otimes I119902) (C otimes I
119902)] [(I
119901otimes B) (I
119901otimes D)]
= [(AC) otimes I119902)] [I119901otimes (BD)]
= (AC) otimes (BD) (6)
Let A[1] = A and define the Kronecker power by
A[119896+1] = A[119896] otimes A = A otimes A[119896] 119896 = 1 2 (7)
FromTheorem 3 we have the following corollary [7]
Corollary 4 If the following matrix products exist then onehas
(1) (A1otimes B1)(A2otimes B2) sdot sdot sdot (A
119901otimes B119901) = (A
1A2sdot sdot sdotA119901) otimes
(B1B2sdot sdot sdotB119901)
(2) (A1otimes A2otimes sdot sdot sdot otimes A
119901)(B1otimes B2otimes sdot sdot sdot otimes B
119901) = (A
1B1) otimes
(A2B2) otimes sdot sdot sdot otimes (A
119901B119901)
(3) [AB][119896]
= A[119896]B[119896]
A squarematrixA is said to be a normalmatrix if and onlyif A119867A = AA119867 A square matrix A is said to be a unitarymatrix if and only if A119867A = AA119867 = I Straightforwardcalculation gives the following conclusions [6 7 33 34]
Theorem 5 For any square matrices A and B(1) if Aminus1 and Bminus1 exist then (A otimes B)
minus1
= Aminus1 otimes Bminus1(2) ifA and B are normal matrices thenAotimesB is a normal
matrix(3) if A and B are unitary (orthogonal) matrices then A otimes
B is a unitary (orthogonal) matrix
Let 120582[A] = 1205821 1205822 120582
119898 denote the eigenvalues of A
and let 120590[A] = 1205901 1205902 120590
119903 denote the nonzero singular
values ofA According to the definition of the eigenvalue andTheorem 3 we have the following conclusions [34]
Theorem 6 Let A isin F119898times119898 and B isin F119899times119899 119896 and 119897 are positiveintegers Then 120582[A119896 otimes B119897] = 120582
119896
119894120583119897
119895| 119894 = 1 2 119898 119895 =
1 2 119899 = 120582[B119897 otimes A119896] Here 120582[A] = 1205821 1205822 120582
119898 and
120582[B] = 1205831 1205832 120583
119899
According to the definition of the singular value andTheorem 3 for any matrices A and B we have the nexttheorem
Theorem 7 Let A isin F119898times119899 and B isin F119901times119902 If rank[A] = 119903120590[A] = 120590
1 1205902 120590
119903 rank[B] = 119904 and 120590[B] =
1205881 1205882 120588
119904 then 120590[A otimes B] = 120590
119894120588119895
| 119894 = 1 2 119903 119895 =
1 2 119904 = 120590[B otimes A]
Proof According to the singular value decomposition theo-rem there exist the unitary matrices U V and W Q whichsatisfy
A = U [Σ 00 0]V B = [
Γ 00 0]Q (8)
where Σ = diag[1205901 1205902 120590
119903] and Γ = diag[120588
1 1205882 120588
119904]
According to Corollary 4 we have
A otimes B = U [Σ 00 0]V otimes W [
Γ 00 0]Q
= (U otimes W) [Σ 00 0] otimes [
Γ 00 0] (V otimes Q)
= (U otimes W) [
[
Σ otimes [Σ 00 0] 00 0
]
]
(V otimes Q)
= (U otimes W) [Σ otimes Γ 00 0] (V otimes Q)
(9)
Since U otimes W and V otimes Q are unitary matrices and Σ otimes Γ =
diag[12059011205881 12059011205882 120590
1120588119904 120590
119903120588119904] this proves the theorem
According toTheorem 7 we have the next corollary
Corollary 8 For any matricesA B andC one has 120590[AotimesBotimes
C] = 120590[C otimes B otimes A]
4 The Properties of the Vector Operator andthe Vec-Permutation Matrix
In this section we introduce a vector-valued operator and avec-permutation matrix
Let A = [a1 a2 a
119899] isin F119898times119899 where a
119895isin F119898 119895 =
1 2 119899 then the vector col[A] is defined by
col [A] =
[[[[
[
a1
a2
a119899
]]]]
]
isin F119898119899
(10)
Theorem 9 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119899 Then
(1) (I119901otimes A)col[B] = col[AB]
(2) (A otimes I119901)col[C] = col[CAT
]
Proof Let (B)119894denote the 119894th column of matrix B we have
(I119901otimes A) col [B] =
[[[[
[
A 0 sdot sdot sdot 00 A sdot sdot sdot 0
d
0 0 sdot sdot sdot A
]]]]
]
[[[[
[
(B)1
(B)2
(B)119901
]]]]
]
=
[[[[
[
A(B)1
A(B)2
A(B)119901
]]]]
]
=
[[[[
[
(AB)1
(AB)2
(AB)119901
]]]]
]
= col [AB]
(11)
4 Journal of Applied Mathematics
Similarly we have
(A otimes I119901) col [C]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
(C)1
(C)2
(C)119899
]]]]
]
=
[[[[
[
11988611(C)1+ 11988612(C)2+ sdot sdot sdot + 119886
1119899(C)119899
11988621(C)1+ 11988622(C)2+ sdot sdot sdot + 119886
2119899(C)119899
1198861198981
(C)1+ 1198861198982
(C)2+ sdot sdot sdot + 119886
119898119899(C)119899
]]]]
]
=
[[[[[
[
C(A119879)1
C(A119879)2
C(A119879)
119898
]]]]]
]
=
[[[[[
[
(CA119879)1
(CA119879)2
(CA119879)
119898
]]]]]
]
= col [CA119879]
(12)
Theorem 10 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119902 Then
col [ABC] = (C119879 otimes A) col [B] (13)
Proof According toTheorems 9 and 1 we have
col [ABC] = col [(AB)C]
= (C119879 otimes I119898) col [AB]
= (C119879 otimes I119898) (I119901otimes A) col [B]
= [(C119879 otimes I119898) (I119901otimes A)] col [B]
= (C119879 otimes A) col [B]
(14)
Theorem 10 plays an important role in solving the matrixequations [25 35ndash37] system identification [38ndash54] andcontrol theory [55ndash58]
Let e119894119899denote an 119899-dimensional column vector which has
1 in the 119894th position and 0rsquos elsewhere that is
e119894119899
= [0 0 0 1 0 0]119879
(15)
Define the vec-permutation matrix
P119898119899
=
[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]
]
isin R119898119899times119898119899
(16)
which can be expressed as [6 7 33 37]119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
(17)
These two definitions of the vec-permutation matrix areequivalent that is
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
= P119898119899
(18)
In fact according to Theorem 3 and the basic properties ofthe Kronecker product we have
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119879119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899e119879119895119898
) otimes (e119895119898e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119879119895119898
) otimes (e119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
[e119896119899
otimes (e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
=
119899
sum119896=1
[
[
e119896119899
otimes
119898
sum119895=1
(e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
]
=
119899
sum119896=1
[e119896119899
otimes I119898
otimes e119879119896119899
]
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= P119898119899
(19)
Based on the definition of the vec-permutationmatrix wehave the following conclusions
Theorem 11 According to the definition of P119898119899 one has
(1) P119879119898119899
= P119899119898
(2) P119879119898119899P119898119899
= P119898119899P119879119898119899
= I119898119899
That is P119898119899
is an (119898119899) times (119898119899) permutation matrix
Journal of Applied Mathematics 5
Proof According to the definition ofP119898119899Theorem 3 and the
basic properties of the Kronecker product we have
P119879119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
119879
= [I119879119898
otimes e1119899 I119879119898
otimes e2119899 I119879
119898otimes e119899119899
]
=
[[[[[[
[
e1198791119898
otimes e1119899
e1198791119898
otimes e2119899
sdot sdot sdot e1198791119898
otimes e119899119899
e1198792119898
otimes e1119899
e1198792119898
otimes e2119899
sdot sdot sdot e1198792119898
otimes e119899119899
e119879119898119898
otimes e1119899
e119879119898119898
otimes e2119899
sdot sdot sdot e119879119898119898
otimes e119899119899
]]]]]]
]
=
[[[[[[
[
e1119899
otimes e1198791119898
e2119899
otimes e1198791119898
sdot sdot sdot e119899119899
otimes e1198791119898
e1119899
otimes e1198792119898
e2119899
otimes e1198792119898
sdot sdot sdot e119899119899
otimes e1198792119898
e1119899
otimes e119879119898119898
e2119899
otimes e119879119898119898
sdot sdot sdot e119899119899
otimes e119879119898119898
]]]]]]
]
=
[[[[[[
[
I119899otimes e1198791119898
I119899otimes e1198792119898
I119899otimes e119879119898119898
]]]]]]
]
= P119899119898
(20)
P119898119899P119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
[I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
=
[[[[[[[
[
I119898
otimes (e1198791119899e1119899) I119898
otimes (e1198791119899e2119899) sdot sdot sdot I
119898otimes (e1198791119899e119899119899
)
I119898
otimes (e1198792119899e1119899) I119898
otimes (e1198792119899e2119899) sdot sdot sdot I
119898otimes (e1198792119899e119899119899
)
I119898
otimes (e119879119899119899e1119899) I119898
otimes (e119879119899119899e2119899) sdot sdot sdot I
119898otimes (e119879119899119899e119899119899
)
]]]]]]]
]
=
[[[[
[
I119898
0 sdot sdot sdot 00 I119898
sdot sdot sdot 0
d
0 0 sdot sdot sdot I119898
]]]]
]
= I119898119899
(21)
P119879119898119899P119898119899
= [I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= I119898
otimes [
119899
sum119894=1
e119894119899e119879119894119899]
= I119898
otimes I119899
= I119898119899
(22)
For any matrix A isin F119898times119899 we have col [A] = P119898119899col [A119879]
Theorem 12 IfA isin F119898times119899 andB isin F119901times119902 then one hasP119898119901
(Aotimes
B)P119879119899119902
= B otimes A
Proof Let B = [119887119894119895] = [
[
B1B2B119901
]
]
where B119894 isin F1times119902 and 119894 =
1 2 119901 and 119895 = 1 2 119902 According to the definition ofP119898119899
and the Kronecker product we have
P119898119901
(A otimes B)P119879119899119902
=
[[[[[[[
[
I119898
otimes e1198791119901
I119898
otimes e1198792119901
I119898
otimes e119879119901119901
]]]]]]]
]
[(A)1otimes B (A)
2otimes B (A)
119899otimes B]P119879
119899119902
=
[[[[
[
(A)1otimes B1 (A)
2otimes B1 sdot sdot sdot (A)
119899otimes B1
(A)1otimes B2 (A)
2otimes B2 sdot sdot sdot (A)
119899otimes B2
(A)1otimes B119901 (A)
2otimes B119901 sdot sdot sdot (A)
119899otimes B119901
]]]]
]
P119879119899119902
=
[[[[
[
A otimes B1A otimes B2
A otimes B119901
]]]]
]
[I119899otimes e1119902 I119899otimes e2119902 I
119899otimes e119902119902]
=
[[[[
[
A11988711
A11988712
sdot sdot sdot A1198871119902
A11988721
A11988722
sdot sdot sdot A1198872119902
A1198871199011
A1198871199012
sdot sdot sdot A119887119901119902
]]]]
]
= B otimes A
(23)
FromTheorem 12 we have the following corollaries
Corollary 13 If A isin F119898times119899 then P119898119903
(A otimes I119903)P119879119899119903
= I119903otimes A
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Similarly we have
(A otimes I119901) col [C]
=
[[[[
[
11988611I119901
11988612I119901
sdot sdot sdot 1198861119899I119901
11988621I119901
11988622I119901
sdot sdot sdot 1198862119899I119901
1198861198981I119901
1198861198982I119901
sdot sdot sdot 119886119898119899I119901
]]]]
]
[[[[
[
(C)1
(C)2
(C)119899
]]]]
]
=
[[[[
[
11988611(C)1+ 11988612(C)2+ sdot sdot sdot + 119886
1119899(C)119899
11988621(C)1+ 11988622(C)2+ sdot sdot sdot + 119886
2119899(C)119899
1198861198981
(C)1+ 1198861198982
(C)2+ sdot sdot sdot + 119886
119898119899(C)119899
]]]]
]
=
[[[[[
[
C(A119879)1
C(A119879)2
C(A119879)
119898
]]]]]
]
=
[[[[[
[
(CA119879)1
(CA119879)2
(CA119879)
119898
]]]]]
]
= col [CA119879]
(12)
Theorem 10 Let A isin F119898times119899 B isin F119899times119901 and C isin F119901times119902 Then
col [ABC] = (C119879 otimes A) col [B] (13)
Proof According toTheorems 9 and 1 we have
col [ABC] = col [(AB)C]
= (C119879 otimes I119898) col [AB]
= (C119879 otimes I119898) (I119901otimes A) col [B]
= [(C119879 otimes I119898) (I119901otimes A)] col [B]
= (C119879 otimes A) col [B]
(14)
Theorem 10 plays an important role in solving the matrixequations [25 35ndash37] system identification [38ndash54] andcontrol theory [55ndash58]
Let e119894119899denote an 119899-dimensional column vector which has
1 in the 119894th position and 0rsquos elsewhere that is
e119894119899
= [0 0 0 1 0 0]119879
(15)
Define the vec-permutation matrix
P119898119899
=
[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]
]
isin R119898119899times119898119899
(16)
which can be expressed as [6 7 33 37]119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
(17)
These two definitions of the vec-permutation matrix areequivalent that is
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
= P119898119899
(18)
In fact according to Theorem 3 and the basic properties ofthe Kronecker product we have
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119895119898
otimes e119896119899
)119879
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119895119898
) (e119879119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899e119879119895119898
) otimes (e119895119898e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
(e119896119899
otimes e119879119895119898
) otimes (e119895119898
otimes e119879119896119899
)
=
119898
sum119895=1
119899
sum119896=1
[e119896119899
otimes (e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
=
119899
sum119896=1
[
[
e119896119899
otimes
119898
sum119895=1
(e119879119895119898
otimes e119895119898
) otimes e119879119896119899
]
]
=
119899
sum119896=1
[e119896119899
otimes I119898
otimes e119879119896119899
]
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= P119898119899
(19)
Based on the definition of the vec-permutationmatrix wehave the following conclusions
Theorem 11 According to the definition of P119898119899 one has
(1) P119879119898119899
= P119899119898
(2) P119879119898119899P119898119899
= P119898119899P119879119898119899
= I119898119899
That is P119898119899
is an (119898119899) times (119898119899) permutation matrix
Journal of Applied Mathematics 5
Proof According to the definition ofP119898119899Theorem 3 and the
basic properties of the Kronecker product we have
P119879119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
119879
= [I119879119898
otimes e1119899 I119879119898
otimes e2119899 I119879
119898otimes e119899119899
]
=
[[[[[[
[
e1198791119898
otimes e1119899
e1198791119898
otimes e2119899
sdot sdot sdot e1198791119898
otimes e119899119899
e1198792119898
otimes e1119899
e1198792119898
otimes e2119899
sdot sdot sdot e1198792119898
otimes e119899119899
e119879119898119898
otimes e1119899
e119879119898119898
otimes e2119899
sdot sdot sdot e119879119898119898
otimes e119899119899
]]]]]]
]
=
[[[[[[
[
e1119899
otimes e1198791119898
e2119899
otimes e1198791119898
sdot sdot sdot e119899119899
otimes e1198791119898
e1119899
otimes e1198792119898
e2119899
otimes e1198792119898
sdot sdot sdot e119899119899
otimes e1198792119898
e1119899
otimes e119879119898119898
e2119899
otimes e119879119898119898
sdot sdot sdot e119899119899
otimes e119879119898119898
]]]]]]
]
=
[[[[[[
[
I119899otimes e1198791119898
I119899otimes e1198792119898
I119899otimes e119879119898119898
]]]]]]
]
= P119899119898
(20)
P119898119899P119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
[I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
=
[[[[[[[
[
I119898
otimes (e1198791119899e1119899) I119898
otimes (e1198791119899e2119899) sdot sdot sdot I
119898otimes (e1198791119899e119899119899
)
I119898
otimes (e1198792119899e1119899) I119898
otimes (e1198792119899e2119899) sdot sdot sdot I
119898otimes (e1198792119899e119899119899
)
I119898
otimes (e119879119899119899e1119899) I119898
otimes (e119879119899119899e2119899) sdot sdot sdot I
119898otimes (e119879119899119899e119899119899
)
]]]]]]]
]
=
[[[[
[
I119898
0 sdot sdot sdot 00 I119898
sdot sdot sdot 0
d
0 0 sdot sdot sdot I119898
]]]]
]
= I119898119899
(21)
P119879119898119899P119898119899
= [I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= I119898
otimes [
119899
sum119894=1
e119894119899e119879119894119899]
= I119898
otimes I119899
= I119898119899
(22)
For any matrix A isin F119898times119899 we have col [A] = P119898119899col [A119879]
Theorem 12 IfA isin F119898times119899 andB isin F119901times119902 then one hasP119898119901
(Aotimes
B)P119879119899119902
= B otimes A
Proof Let B = [119887119894119895] = [
[
B1B2B119901
]
]
where B119894 isin F1times119902 and 119894 =
1 2 119901 and 119895 = 1 2 119902 According to the definition ofP119898119899
and the Kronecker product we have
P119898119901
(A otimes B)P119879119899119902
=
[[[[[[[
[
I119898
otimes e1198791119901
I119898
otimes e1198792119901
I119898
otimes e119879119901119901
]]]]]]]
]
[(A)1otimes B (A)
2otimes B (A)
119899otimes B]P119879
119899119902
=
[[[[
[
(A)1otimes B1 (A)
2otimes B1 sdot sdot sdot (A)
119899otimes B1
(A)1otimes B2 (A)
2otimes B2 sdot sdot sdot (A)
119899otimes B2
(A)1otimes B119901 (A)
2otimes B119901 sdot sdot sdot (A)
119899otimes B119901
]]]]
]
P119879119899119902
=
[[[[
[
A otimes B1A otimes B2
A otimes B119901
]]]]
]
[I119899otimes e1119902 I119899otimes e2119902 I
119899otimes e119902119902]
=
[[[[
[
A11988711
A11988712
sdot sdot sdot A1198871119902
A11988721
A11988722
sdot sdot sdot A1198872119902
A1198871199011
A1198871199012
sdot sdot sdot A119887119901119902
]]]]
]
= B otimes A
(23)
FromTheorem 12 we have the following corollaries
Corollary 13 If A isin F119898times119899 then P119898119903
(A otimes I119903)P119879119899119903
= I119903otimes A
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Proof According to the definition ofP119898119899Theorem 3 and the
basic properties of the Kronecker product we have
P119879119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
119879
= [I119879119898
otimes e1119899 I119879119898
otimes e2119899 I119879
119898otimes e119899119899
]
=
[[[[[[
[
e1198791119898
otimes e1119899
e1198791119898
otimes e2119899
sdot sdot sdot e1198791119898
otimes e119899119899
e1198792119898
otimes e1119899
e1198792119898
otimes e2119899
sdot sdot sdot e1198792119898
otimes e119899119899
e119879119898119898
otimes e1119899
e119879119898119898
otimes e2119899
sdot sdot sdot e119879119898119898
otimes e119899119899
]]]]]]
]
=
[[[[[[
[
e1119899
otimes e1198791119898
e2119899
otimes e1198791119898
sdot sdot sdot e119899119899
otimes e1198791119898
e1119899
otimes e1198792119898
e2119899
otimes e1198792119898
sdot sdot sdot e119899119899
otimes e1198792119898
e1119899
otimes e119879119898119898
e2119899
otimes e119879119898119898
sdot sdot sdot e119899119899
otimes e119879119898119898
]]]]]]
]
=
[[[[[[
[
I119899otimes e1198791119898
I119899otimes e1198792119898
I119899otimes e119879119898119898
]]]]]]
]
= P119899119898
(20)
P119898119899P119898119899
=
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
[I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
=
[[[[[[[
[
I119898
otimes (e1198791119899e1119899) I119898
otimes (e1198791119899e2119899) sdot sdot sdot I
119898otimes (e1198791119899e119899119899
)
I119898
otimes (e1198792119899e1119899) I119898
otimes (e1198792119899e2119899) sdot sdot sdot I
119898otimes (e1198792119899e119899119899
)
I119898
otimes (e119879119899119899e1119899) I119898
otimes (e119879119899119899e2119899) sdot sdot sdot I
119898otimes (e119879119899119899e119899119899
)
]]]]]]]
]
=
[[[[
[
I119898
0 sdot sdot sdot 00 I119898
sdot sdot sdot 0
d
0 0 sdot sdot sdot I119898
]]]]
]
= I119898119899
(21)
P119879119898119899P119898119899
= [I119898
otimes e1119899 I119898
otimes e2119899 I
119898otimes e119899119899
]
[[[[[[
[
I119898
otimes e1198791119899
I119898
otimes e1198792119899
I119898
otimes e119879119899119899
]]]]]]
]
= I119898
otimes [
119899
sum119894=1
e119894119899e119879119894119899]
= I119898
otimes I119899
= I119898119899
(22)
For any matrix A isin F119898times119899 we have col [A] = P119898119899col [A119879]
Theorem 12 IfA isin F119898times119899 andB isin F119901times119902 then one hasP119898119901
(Aotimes
B)P119879119899119902
= B otimes A
Proof Let B = [119887119894119895] = [
[
B1B2B119901
]
]
where B119894 isin F1times119902 and 119894 =
1 2 119901 and 119895 = 1 2 119902 According to the definition ofP119898119899
and the Kronecker product we have
P119898119901
(A otimes B)P119879119899119902
=
[[[[[[[
[
I119898
otimes e1198791119901
I119898
otimes e1198792119901
I119898
otimes e119879119901119901
]]]]]]]
]
[(A)1otimes B (A)
2otimes B (A)
119899otimes B]P119879
119899119902
=
[[[[
[
(A)1otimes B1 (A)
2otimes B1 sdot sdot sdot (A)
119899otimes B1
(A)1otimes B2 (A)
2otimes B2 sdot sdot sdot (A)
119899otimes B2
(A)1otimes B119901 (A)
2otimes B119901 sdot sdot sdot (A)
119899otimes B119901
]]]]
]
P119879119899119902
=
[[[[
[
A otimes B1A otimes B2
A otimes B119901
]]]]
]
[I119899otimes e1119902 I119899otimes e2119902 I
119899otimes e119902119902]
=
[[[[
[
A11988711
A11988712
sdot sdot sdot A1198871119902
A11988721
A11988722
sdot sdot sdot A1198872119902
A1198871199011
A1198871199012
sdot sdot sdot A119887119901119902
]]]]
]
= B otimes A
(23)
FromTheorem 12 we have the following corollaries
Corollary 13 If A isin F119898times119899 then P119898119903
(A otimes I119903)P119879119899119903
= I119903otimes A
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Corollary 14 If A isin F119898times119899 and B isin F119899times119898 then
B otimes A = P119898119899
(A otimes B)P119879119899119898
= P119898119899
[(A otimes B)P2119898119899
]P119879119899119898
(24)
That is 120582[B otimes A] = 120582[(A otimes B)P2119898119899
] When A isin F119899times119899 andB isin F 119905times119905 one has B otimes A = P
119899119905(A otimes B)P119879
119899119905 That is if A and B
are square matrices then A otimes B is similar to B otimes A
5 The Scalar Properties and the PolynomialsMatrix of the Kronecker Product
In this section we discuss the properties [6 7 34] of thedeterminant the trace the rank and the polynomial matrixof the Kronecker product
ForA isin F119898times119898 andB isin F119899times119899 we have |AotimesB| = |A|119899
|B|119898
=
|B otimes A| If A and B are two square matrices then we havetr[A otimes B] = tr[A] tr[B] = tr[B otimes A] For any matrices A andB we have rank[A otimes B] = rank[A] rank[B] = rank[B otimes A]According to these scalar properties we have the followingtheorems
Theorem 15 (1) Let AC isin F119898times119898 and BD isin F119899times119899 Then
|(A otimes B) (C otimes D)| = |(A otimes B)| |(C otimes D)|
= (|A| |C|)119899
(|B||D|)119898
= |AC|119899
|BD|119898
(25)
(2) If A B C and D are square matrices then
tr [(A otimes B) (C otimes D)] = tr [(AC) otimes (BD)]
= tr [AC] tr [BD]
= tr [CA] tr [DB]
(26)
(3) Let A isin F119898times119899 C isin F119899times119901 B isin F119902times119903 and D isin F 119903times119904 then
rank [(A otimes B) (C otimes D)] = rank [(AC) otimes (BD)]
= rank [AC] rank [BD] (27)
Theorem 16 If 119891(119909 119910) = 119909119903
119910119904 is a monomial and 119891(AB) =
A[119903]otimesB[119904] where 119903 119904 are positive integers one has the followingconclusions
(1) Let A isin F119898times119898 and B isin F119899times119899 Then
1003816100381610038161003816119891 (AB)1003816100381610038161003816 = |A|
119903119898119903minus1119899119904
|B|119904119898119903119899119904minus1
(28)
(2) If A and B are square matrices then
tr [119891 (AB)] = 119891 (tr [A] tr [B]) (29)
(3) For any matrices A and B one has
rank [119891 (AB)] = 119891 (rank [A] rank [B]) (30)
If 120582[A] = 1205821 1205822 120582
119898 and 119891(119909) = sum
119896
119894=1119888119894119909119894 is a
polynomial then the eigenvalues of
119891 (A) =
119896
sum119894=1
119888119894A119894 (31)
are
119891 (120582119895) =
119896
sum119894=1
119888119894120582119894
119895 119895 = 1 2 119898 (32)
Similarly consider a polynomial 119891(119909 119910) in two variables 119909
and 119910
119891 (119909 119910) =
119896
sum119894119895=1
119888119894119895119909119894
119910119895
119888119894119895 119909 119910 isin F (33)
where 119896 is a positive integer Define the polynomial matrix119891(AB) by the formula
119891 (AB) =
119896
sum119894119895=1
119888119894119895A119894 otimes B119895 (34)
According to Theorem 3 we have the following theorems[34]
Theorem 17 Let A isin F119898times119898 and B isin F119899times119899 if 120582[A] =
1205821 1205822 120582
119898 and 120582[B] = 120583
1 1205832 120583
119899 then the matrix
119891(AB) has the eigenvalues
119891 (120582119903 120583119904) =
119896
sum119894119895=1
119888119894119895120582119894
119903120583119895
119904 119903 = 1 2 119898 119904 = 1 2 119899
(35)
Theorem 18 (see [34]) Let A isin F119898times119898 If 119891(119911) is an analyticfunction and 119891(A) exists then
119891(I119899otimes A) = I
119899otimes 119891(A)
119891(A otimes I119899) = 119891(A) otimes I
119899
Finally we introduce some results about the Kroneckersum [7 34] The Kronecker sum of A isin F119898times119898 and B isin F119899times119899denoted as A oplus B is defined by
A oplus B = A otimes I119899+ I119898
otimes B
Theorem 19 Let A isin F119898times119898 and B isin F119899times119899 Then
exp[A oplus B] = exp[A] otimes exp[B]sin(A oplus B) = sin(A) otimes cos(B) + cos(A) otimes sin(B)cos(A oplus B) = cos(A) otimes cos(B) minus sin(A) otimes sin(B)
6 Conclusions
This paper establishes some conclusions on the Kroneckerproducts and the vec-permutation matrix A new presen-tation about the properties of the mixed products and thevector operator is given All these obtained conclusions makethe theory of the Kronecker product more complete
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (no 61273194) the 111 Project (B12018)and the PAPD of Jiangsu Higher Education Institutions
References
[1] HV Jemderson F Pukelsheim and S R Searle ldquoOn the historyof the Kronecker productrdquo Linear and Multilinear Algebra vol14 no 2 pp 113ndash120 1983
[2] X L Xiong W Fan and R Ding ldquoLeast-squares parameterestimation algorithm for a class of input nonlinear systemsrdquoJournal of Applied Mathematics vol 2007 Article ID 68407414 pages 2007
[3] F Ding ldquoTransformations between some special matricesrdquoComputers amp Mathematics with Applications vol 59 no 8 pp2676ndash2695 2010
[4] Y Shi and B Yu ldquoOutput feedback stabilization of networkedcontrol systems with random delays modeled by Markovchainsrdquo IEEE Transactions on Automatic Control vol 54 no 7pp 1668ndash1674 2009
[5] Y Shi H Fang and M Yan ldquoKalman filter-based adaptivecontrol for networked systems with unknown parameters andrandomly missing outputsrdquo International Journal of Robust andNonlinear Control vol 19 no 18 pp 1976ndash1992 2009
[6] A Graham Kronecker Products and Matrix Calculus WithApplications John Wiley amp Sons New York NY USA 1982
[7] W-H Steeb and Y Hardy Matrix Calculus and KroneckerProduct A Practical Approach to Linear andMultilinear AlgebraWorld Scientific River Edge NJ USA 2011
[8] P M Bentler and S Y Lee ldquoMatrix derivatives with chainrule and rules for simple Hadamard and Kronecker productsrdquoJournal of Mathematical Psychology vol 17 no 3 pp 255ndash2621978
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications to simple Hadamard and Kronecker prod-uctsrdquo Journal ofMathematical Psychology vol 29 no 4 pp 474ndash492 1985
[10] F Ding and T Chen ldquoIterative least-squares solutions ofcoupled Sylvester matrix equationsrdquo Systems amp Control Lettersvol 54 no 2 pp 95ndash107 2005
[11] F Ding and T Chen ldquoOn iterative solutions of general coupledmatrix equationsrdquo SIAM Journal on Control and Optimizationvol 44 no 6 pp 2269ndash2284 2006
[12] L Jodar andHAbou-Kandil ldquoKronecker products and coupledmatrix Riccati differential systemsrdquo Linear Algebra and itsApplications vol 121 no 2-3 pp 39ndash51 1989
[13] D Bahuguna A Ujlayan and D N Pandey ldquoAdvanced typecoupled matrix Riccati differential equation systems with Kro-necker productrdquo Applied Mathematics and Computation vol194 no 1 pp 46ndash53 2007
[14] M Dehghan and M Hajarian ldquoAn iterative algorithm forsolving a pair of matrix equations 119860119884119861 = 119864 119862119884119863 = 119865
over generalized centro-symmetric matricesrdquo Computers ampMathematics with Applications vol 56 no 12 pp 3246ndash32602008
[15] M Dehghan and M Hajarian ldquoAn iterative algorithm for thereflexive solutions of the generalized coupled Sylvester matrixequations and its optimal approximationrdquoAppliedMathematicsand Computation vol 202 no 2 pp 571ndash588 2008
[16] C F van Loan ldquoThe ubiquitous Kronecker productrdquo Journal ofComputational and Applied Mathematics vol 123 no 1-2 pp85ndash100 2000
[17] M Huhtanen ldquoReal linear Kronecker product operationsrdquoLinear Algebra and its Applications vol 418 no 1 pp 347ndash3612006
[18] S Delvaux and M van Barel ldquoRank-deficient submatrices ofKronecker products of Fourier matricesrdquo Linear Algebra and itsApplications vol 426 no 2-3 pp 349ndash367 2007
[19] S G Deo K N Murty and J Turner ldquoQualitative properties ofadjoint Kronecker product boundary value problemsrdquo AppliedMathematics and Computation vol 133 no 2-3 pp 287ndash2952002
[20] W-H Steeb and F Wilhelm ldquoExponential functions of Kro-necker products and trace calculationrdquo Linear and MultilinearAlgebra vol 9 no 4 pp 345ndash346 1981
[21] J Chuai and Y Tian ldquoRank equalities and inequalities forKronecker products of matrices with applicationsrdquo AppliedMathematics and Computation vol 150 no 1 pp 129ndash137 2004
[22] R H Koning H Neudecker and T Wansbeek ldquoBlock Kro-necker products and the vecb operatorrdquo Linear Algebra and itsApplications vol 149 pp 165ndash184 1991
[23] F Ding P X Liu and J Ding ldquoIterative solutions of thegeneralized Sylvester matrix equations by using the hierarchicalidentification principlerdquo Applied Mathematics and Computa-tion vol 197 no 1 pp 41ndash50 2008
[24] L Xie Y Liu and H Yang ldquoGradient based and least squaresbased iterative algorithms for matrix equations119860119883119861+119862119883
T119863 =
119865rdquo Applied Mathematics and Computation vol 217 no 5 pp2191ndash2199 2010
[25] F Ding and T Chen ldquoGradient based iterative algorithmsfor solving a class of matrix equationsrdquo IEEE Transactions onAutomatic Control vol 50 no 8 pp 1216ndash1221 2005
[26] J Ding Y Liu and F Ding ldquoIterative solutions to matrixequations of the form 119860
119894119883119861119894= 119865119894rdquo Computers amp Mathematics
with Applications vol 59 no 11 pp 3500ndash3507 2010[27] A-G Wu L Lv and G-R Duan ldquoIterative algorithms for
solving a class of complex conjugate and transpose matrixequationsrdquo Applied Mathematics and Computation vol 217 no21 pp 8343ndash8353 2011
[28] A-G Wu X Zeng G-R Duan and W-J Wu ldquoIterative sol-utions to the extended Sylvester-conjugate matrix equationsrdquoApplied Mathematics and Computation vol 217 no 1 pp 130ndash142 2010
[29] F Zhang Y Li W Guo and J Zhao ldquoLeast squares solutionswith special structure to the linear matrix equation 119860119883119861 =
119862rdquo Applied Mathematics and Computation vol 217 no 24 pp10049ndash10057 2011
[30] M Dehghan and M Hajarian ldquoSSHI methods for solving gen-eral linearmatrix equationsrdquo Engineering Computations vol 28no 8 pp 1028ndash1043 2011
[31] E Erkmen and M A Bradford ldquoCoupling of finite elementand meshfree methods be for locking-free analysis of shear-deformable beams and platesrdquo Engineering Computations vol28 no 8 pp 1003ndash1027 2011
[32] A Kaveh and B Alinejad ldquoEigensolution of Laplacian matricesfor graph partitioning and domain decomposition approximatealgebraic methodrdquo Engineering Computations vol 26 no 7 pp828ndash842 2009
[33] X Z ZhanTheTheory of Matrces Higher Education Press Bei-jing China 2008 (Chinese)
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
[34] P Lancaster and M Tismenetsky The Theory of Matrices withApplications Academic Press New York NY USA 1985
[35] M Dehghan and M Hajarian ldquoAn iterative method for solvingthe generalized coupled Sylvester matrix equations over gener-alized bisymmetric matricesrdquo Applied Mathematical Modellingvol 34 no 3 pp 639ndash654 2010
[36] M Dehghan and M Hajarian ldquoAn efficient algorithm for solv-ing general coupled matrix equations and its applicationrdquoMathematical and Computer Modelling vol 51 no 9-10 pp1118ndash1134 2010
[37] N J Higham Accuracy and Stability of Numerical AlgorithmsSociety for Industrial and Applied Mathematics PhiladelphiaPa USA 1996
[38] F Ding ldquoDecomposition based fast least squares algorithm foroutput error systemsrdquo Signal Processing vol 93 no 5 pp 1235ndash1242 2013
[39] F Ding ldquoCoupled-least-squares identification for multivariablesystemsrdquo IET Control Theory and Applications vol 7 no 1 pp68ndash79 2013
[40] F Ding X G Liu and J Chu ldquoGradient-based and least-squares-based iterative algorithms for Hammerstein systemsusing the hierarchical identification principlerdquo IET ControlTheory and Applications vol 7 pp 176ndash184 2013
[41] F Ding ldquoHierarchical multi-innovation stochastic gradientalgorithm for Hammerstein nonlinear system modelingrdquoApplied Mathematical Modelling vol 37 no 4 pp 1694ndash17042013
[42] F Ding ldquoTwo-stage least squares based iterative estima-tion algorithm for CARARMA system modelingrdquo AppliedMathemat- Ical Modelling vol 37 no 7 pp 4798ndash4808 2013
[43] Y J Liu Y S Xiao and X L Zhao ldquoMulti-innovation stochasticgradient algorithm for multiple-input single-output systemsusing the auxiliary modelrdquo Applied Mathematics and Compu-tation vol 215 no 4 pp 1477ndash1483 2009
[44] Y J Liu J Sheng and R F Ding ldquoConvergence of stochasticgradient estimation algorithm for multivariable ARX-like sys-temsrdquo Computers amp Mathematics with Applications vol 59 no8 pp 2615ndash2627 2010
[45] J H Li ldquoParameter estimation for Hammerstein CARARMAsystems based on the Newton iterationrdquo Applied MathematicsLetters vol 26 no 1 pp 91ndash96 2013
[46] J H Li R F Ding and Y Yang ldquoIterative parameter identifi-cation methods for nonlinear functionsrdquo Applied MathematicalModelling vol 36 no 6 pp 2739ndash2750 2012
[47] J Ding F Ding X P Liu andG Liu ldquoHierarchical least squaresidentification for linear SISO systems with dual-rate sampled-datardquo IEEE Transactions on Automatic Control vol 56 no 11pp 2677ndash2683 2011
[48] J Ding and F Ding ldquoBias compensation-based parameter esti-mation for output error moving average systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 25 no 12pp 1100ndash1111 2011
[49] J Ding L L Han and X M Chen ldquoTime series AR modelingwithmissing observations based on the polynomial transforma-tionrdquoMathematical andComputerModelling vol 51 no 5-6 pp527ndash536 2010
[50] F Ding Y J Liu and B Bao ldquoGradient-based and least-squares-based iterative estimation algorithms for multi-input multi-output systemsrdquo Proceedings of the Institution of MechanicalEngineers I vol 226 no 1 pp 43ndash55 2012
[51] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based least-squares identification algorithm for one-step state-delay systemsrdquo International Journal of Computer Mathematicsvol 89 no 15 pp 2019ndash2028 2012
[52] FDing andYGu ldquoPerformance analysis of the auxiliarymodel-based stochastic gradient parameter estimation algorithm forstate space systems with one-step state delayrdquo Circuits Systemsand Signal Processing vol 32 no 2 pp 585ndash599 2013
[53] F Ding andHHDuan ldquoTwo-stage parameter estimation algo-rithms for Box-Jenkins systemsrdquo IET Signal Processing 2013
[54] P P Hu and F Ding ldquoMultistage least squares based iterativeestimation for feedback nonlinear systems withmoving averagenoises using the hierarchical identification principlerdquoNonlinearDynamics 2013
[55] H G Zhang and X P Xie ldquoRelaxed stability conditions forcontinuous-time TS fuzzy-control systems via augmentedmulti-indexed matrix approachrdquo IEEE Transactions on FuzzySystems vol 19 no 3 pp 478ndash492 2011
[56] H G Zhang D W Gong B Chen and Z W Liu ldquoSyn-chronization for coupled neural networks with interval delaya novel augmented Lyapunov-Krasovskii functional methodrdquoIEEE Transactions on Neural Networks and Learning Systemsvol 24 no 1 pp 58ndash70 2013
[57] H W Yu and Y F Zheng ldquoDynamic behavior of multi-agentsystems with distributed sampled controlrdquo Acta AutomaticaSinica vol 38 no 3 pp 357ndash363 2012
[58] Q Z Huang ldquoConsensus analysis of multi-agent discrete-timesystemsrdquo Acta Automatica Sinica vol 38 no 7 pp 1127ndash11332012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of