An m n matrix is an rectangular array of elements with m rows and n columns:
mnm
n
aa
aa
A
1
111
Matrices
ija
denotes the element in the ith row and jth column
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4/122
Xe
i
34010
301
242
413
X
t
t
et
te
)3sin(
)3cos(2
FE
DCA
Partitioning in Partitioning in submatricessubmatrices
20101
41302
51010
16301
32242
40413
Matrices y vectores son Matrices y vectores son fundamentales en el fundamentales en el
estudio formal de todas las estudio formal de todas las ramas de la ingenieríaramas de la ingeniería
InstrumentaciónInstrumentaciónDiseDiseño de circuitosño de circuitosComunicacionesComunicacionesMicroelectrónicaMicroelectrónica
A column vector is a matrix with n rows and 1 column
Vectors
na
a
a 1
A row vector is a matrix with 1 row and n columns
naaa ... 1
Square:
3
200
010
342
nm
A
mxn
Classification of matrices
m=n
Symmetric:
203
014
342
A
aji = aij
Upper Triangular:
200
010
342
Aaij = 0 when j < i
333231
232221
131211
aaa
aaa
aaa
A
Lower Triangular:
203
004
002
Aaij = 0 when j >i
333231
232221
131211
aaa
aaa
aaa
A
Diagonal:
200
010
002
Aaij = 0 when j i
Identity:
100
010
001
Aaii = 1
aij = 0 when j i
Sum of matrices of the same dimension:
mnmnmm
mm
baba
baba
BA
11
111111
Scalar multiplicationScalar multiplication BB = = kkAA
Dimensions: Dimensions:
ExampleExample
33
96
11
323
Matrix multiplicationMatrix multiplication
CC = = ABAB
Only possible if the number of Only possible if the number of columns of columns of AA is equal to the is equal to the number of rows of number of rows of BB
312321221121
3
11221
bababa
baci
ii
rkmjbacn
iikjijk ,,1,,1
1
BAC
3
2
1
4
231
220
848
012
310
212
101
321
examples:
Matrix multiplicationMatrix multiplicationis a non-commutative operation is a non-commutative operation
(generally) ::BAAB
Identity:
100
010
001
Aaii = 1
aij = 0 when j i
AIAAII
10
01
Vector products: Vector products: (u,v are column vectors)(u,v are column vectors)
Dot product or Dot product or inner productinner product
Outer product:Outer product:
22112
121 vuvu
v
vuuvuT
2212
211121
2
1
vuvu
vuvuvv
u
uvu T
2
22112
121 uuuuu
u
uuuuuT
Scalar product (of vectors)
The product of a row vector a and a column vector b is a scalar
a b = a1b1 + ... + anbn
n
iiibaba
1
cosbaba
vectorsorthogonalba 0
aaa
TraceTrace
The trace of a nxn matrix A is given by:The trace of a nxn matrix A is given by:
nn
n
iii aaaaATrace
...)( 22111
Properties of Matrix Properties of Matrix OperationsOperations
a)a) A+B = B+AA+B = B+A
b)b) A+(B+C) = A+(B+C) = (A+B)+C(A+B)+C
c)c) A(BC) = (AB)CA(BC) = (AB)C
d)d) A(B+C) = A(B+C) = AB+ACAB+AC
e)e) (B+C)A = (B+C)A = BA+CABA+CA
f)f) a(B+C) = a(B+C) = aB+aCaB+aC
Commutative law for Commutative law for additionaddition
Associative law for Associative law for additionaddition
Associative for Associative for multiplicationmultiplication
Left distributive lawLeft distributive law
Right distributive lawRight distributive lawDistributive law for Distributive law for
scalar multiplicationscalar multiplication
j)j) (a+b)C = aC+bC(a+b)C = aC+bCk)k) a(bC) = (ab)Ca(bC) = (ab)Cl)l) a(BC) = (aB)Ca(BC) = (aB)C
TransposeTranspose
BB = = AATT
Dimensions:Dimensions:
Formula:Formula:
ExampleExample
ijjiij bBab
642
531
65
43
21T
Alternative notation used in Alternative notation used in some bookssome books
BB = = AATT
BB = = AA’’
In this course we use the first one (B = AT )
Transpose Matrix Transpose Matrix propertiesproperties
TTT
TT
TTT
TT
cc
ABAB
AABABA
AA
)(
)(
Symmetric matrix: Symmetric matrix: AATT = = AA
Skew-symmetric matrix: Skew-symmetric matrix: AATT = - = -AA
Unitary matrix example :Unitary matrix example :
21
21
21
21
A
SymmetricSymmetric
Skew-symmetricSkew-symmetric
Unitary matrixUnitary matrix
Given any matrix A with real entries:Given any matrix A with real entries:
symmetricskewisAA
symmetricisAAT
T
Complex conjugate of Complex conjugate of matricesmatrices
Alternative notation used in some books Alternative notation used in some books for Matrix Complex Conjugatefor Matrix Complex Conjugate
In this notes we use the bar *A
A
Complex HermitianComplex Hermitian
TH AA
TH AA
____
ii
i
i
iiH
521
643
526
143
Example:
Complex Hermitian Complex Hermitian PropertiesProperties
HHH
HH
HHH
HH
cc
ABAB
AA
BABA
AA
)(
definitionsdefinitions
examples:examples:
Hermitian:
Skew-Hermitian
Unitary
Given any matrix A with complex Given any matrix A with complex entries:entries:
MatrixhermitianSkewaisAA
MatrixHermitiananisAAT
T
(a) Find A such as:
10
653
02
1132 AAT
(b) Find A such as:
21
02
01
103 TAA
Exercises :
21
02
01
103 TAA
02
213 TAA
dc
baAGiven
02
21
33
33
db
ca
dc
ba
02
21
23
32
dbc
cba
021
21
21
A
ExercisesExercises
Ejercicio: Ejercicio: SimplificarSimplificar
TTT ABCX
TTT ABCX
TTTT ABCX
TTT ABCX
TBACX
Ejercicio: Ejercicio: SimplificarSimplificar TTT ABABY 1
TTTT BAABY 1
BAABYTT 1
BAABYTT 1
BAABYT1
BY 2