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Graduate Studies and Research
Masters Theses & Specialist Projects
Western Kentucky University Year 2009
Qualitative Behavior of Solutions to
Differential Equations in Rn and in
Hilbert SpaceQian Dong
Western Kentucky University, [email protected]
This paper is posted at TopSCHOLAR.
http://digitalcommons.wku.edu/theses/59
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QUALITATIVE BEHAVIOR OF SOLUTIONS TO
DIFFERENTIAL EQUATIONS IN nR AND IN HILBERT SPACE
A Thesis
Presented to
The Faculty of the Department of Mathematics
Western Kentucky University
Bowling Green, Kentucky
In Partial Fulfillment
Of the Requirements for the Degree
Master of Science
By
Qian Dong
May 2009
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QUALITATIVE BEHAVIOR OF SOLUTIONS TO
DIFFERENTIAL EQUATIONS IN nR AND IN HILBERT SPACE
Date Recommended 04/30/2009________
___ Lan Nguyen _________________________Director of Thesis
___ John Spraker _________________________
____ Di Wu ____________________________
_________________________________________Dean, Graduate Studies and Research Date
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iii
ACKNOWLEDGMENTS
My sincere thanks go to Dr. Lan Nguyen, for his guidance and flexibility for the
duration of my thesis project. His excellent mentorship has helped me in understanding
the concepts of qualitative behavior of solutions to differential equations which is the
foundation of my thesis. I just want to say, without him, this thesis would not have been
possible.
Also, I would like to extend my sincere thanks to Dr. John Spraker and Dr. Di Wu
for their service as members of my thesis committee and for their valuable suggestions on
my thesis.
My special thanks go out to my parents who are in China. Without their
encouragement, I would not achieve my goals.
Last but not least, I would like to thank the entire graduate faculty in the
mathematics department at Western Kentucky University for making my graduate
experience such a positive one.
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iv
TABLE OF CONTENTS
ABSTRACT........................................................................................................................ vPREFACE........................................................................................................................... 1CHAPTER 1: Background: A Population Problem........................................................ 41.1 When the solution to population equation is periodic .................................................. 5
1.2 The initial value of the periodic function...... 6
1.3(a) Fish in a lake............................................................................................................. 71.3(b) Population in a village.............................................................................................. 7CHAPTER 2: Matrices ..................................................................................................... 102.1 Space nR , nn Matrices and Their Properties.......................................................... 102.2 Derivative of n-dimensional function......................................................................... 142. 3 Eigenvalues and Eigenvectors of a Matrix ................................................................ 152.4 Matrix Exponential Function
tAe ................................................................................ 16
2.5 The Cauchy Integral Formula of Exponential Function ............................................. 242.6 Additional Functions................................................................................................... 302.7 Spectral Mapping Theorem......................................................................................... 31CHAPTER 3: Qualitative Behavior of Solutions of Differential Equations .................... 34Theorem 3.1 (Existence and Uniqueness Theorem)......................................................... 34Theorem 3.2 (Lyapunovs Theorem)................................................................................ 35Corollary 3.3(Boundedness of solutions of Non-homogeneous DE) ............................... 39Theorem 3.4 (Periodicity Function).................................................................................. 41Theorem 3.5 (Existence Periodic Solution)...................................................................... 42Theorem 3.7 (Boundedness of the complete trajectory) ................................................... 45CHAPTER 4: Extension of Results to Hilbert Spaces..................................................... 461. Hilbert Space and its Properties.................................................................................... 462. The Spectrum of Operator ............................................................................................ 493. Spectral Mapping Theorem in Hilbert Space................................................................ 514. Extension of the Main Results to Hilbert Space ........................................................... 53BIBLIOGRAPHY.60
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v
QUALITATIVE BEHAVIOR OF SOLUTIONS TO
DIFFERENTIAL EQUATIONS IN nR AND IN HILBERT SPACE
Qian Dong May 2009 60 Pages
Directed by Dr. Lan Nguyen
Department of Mathematics Western Kentucky University
ABSTRACT
The qualitative behavior of solutions of differential equations mainly addresses the
various questions arising in the study of the long run behavior of solutions. The contents
of this thesis are related to three of the major problems of the qualitative theory, namely
the stability, the boundedness and the periodicity of the solution. Learning the qualitative
behavior of such solutions is crucial part of the theory of differential equations. It is
important to know if a solution is bounded or unbounded or if a solution is stable, i.e.
0)(lim =
tut
. Moreover, the periodicity of a solution is also of great significance for
practical purposes.
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1
PREFACE
In mathematics, different processes can be combined to help us prove a more
comprehensive result. In fact, it is almost certain that when solving new problems, we use
more knowledge that we have already acquired to reach a new conclusion. The
qualitative behavior of solutions to differential equations mainly addresses various
questions arising in the study of the long run behavior of solutions. The contents of this
thesis are related to three of the major problems of the qualitative theory, namely
stability, boundedness and periodicity of the solution.
It is our view that one of the most important problems in the study of homogeneous
and non-homogeneous equations and their applications is that of describing the nature of
the solutions for a large range of parameters involved. From a numerical point of view,
the existence of a periodic solution of the population equations approximation scheme
must also be studied. The usual approach to fulfill such requirements is to have a set of
differential equations which are as general as possible and for which explicit analytic
conditions can be given.
Below, we are going to explain how to find the qualitative behavior of solutions to
differential equations in nR in three main chapters.
In Chapter 1, we analyze the non-homogeneous differential equation in 1-
dimensional R with periodic solutions, then give applications of the asymptotic behavior
of solutions of the ordinary differential equations inR with periodic solution in the real
world and studying periodic solution of a population equation that represents real-world
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situations. We will also achieve some results for the population equation in 1-
dimensionalR as a good beginning of the multi-dimensional case. In this context, most of
the attention has been given to one periodic solution in 1- dimensionalR . A periodicsolution with initial population 0y ensures that the population cannot become extinct,
provided )()1( tyty =+ .
It is important to study not only in 1- dimension, but in multi-dimensional linear
equations. Most obvious applications would be in the studies of Linear Algebra and
Differential Equations where matrix functions are prevalent. To reach our final results,
we are going to study space nR , nn matrices and their properties. In Chapter 2, we
introduce a matrix-valued exponential function and properties of such exponential
function. Using Riesz theory, we also introduce the matrix-valued function )(Af , where
)(zf is a given analytic function and A is a square matrix. If we look at zezf =)( , an
exponential function, then we can define matrix Ae . Many properties of such functions
are given. They are very important to the theory of matrix-valued differential equations
and the behavior of their solutions. At the end of Chapter 3, we prove the Spectral
Mapping Theorem, an exemplary theorem about the relationship between the eigenvalue
set of a matrix A and the eigenvalue set of matrix )(Af .
Finally, we have the main results in Chapter 3 and Chapter 4. Learning the
qualitative behavior of such solutions is an important part of the theory of differential
equations. Namely given the system:
=
+=
0)0(
)()()('
yy
tftAyty
,
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where A is a linear operator in a Hilbert space, it is important to know if a solution is
bounded or unbounded, or if a solution is stable, i.e.0)(lim =
tu
t
. Moreover, the
periodicity of a solution is also of great significance for practical purposes. Among the
results, we have the theorem about the stability of solutions of homogeneous equation. It
gives four equivalent conditions to check if the system is stable. As a nice corollary of
that result, if 0)Re(
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CHAPTER 1:
Background: A Population Problem
In this chapter, we consider a population (such as human beings, bacteria, fish, etc.)
model. In this population, we assume the birth rate = b, the death rate = d, then the
growth rate: dbr = . If we have external influence then each year )(tf is added (or
subtracted) to the population.
Let )(ty be the population at time t. Then, we have the population equation:
=
+=
0)0(
)()()('
yy
tftryty(1)
We should use the method for solving linear differential equations. First, write the
equation in the standard form:
)()()(' tftryty = (2)
Using the integrating factor:
rtrdt eet ==)( (3)
We have the solution:
+=
tstrrt dssfeyety
0
)(0 )()( (4)
We consider the following question: Is )(ty periodic if )(tf is periodic? Recall, a
function )(tf is calledp-periodic if )()( tfptf =+ for all tin the domain. For the sake
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of simplicity we choose 1=p . If )(tf is 1-periodic then, in general, the solution )(ty is
not periodic. We now want to find certain initial value 0y , so that y(t) is periodic. We
now are in the position to find the initial value, such that the solution )(ty is periodic.
1.1 When the solution to population equation is periodic
Theorem 1.1 Suppose )(tf is a periodic function with period 1. If 0r , then there exists
a unique initial value 0y , such that the solution of the population equation:
=
+=
0)0(
)()()('
yy
tftryty
is 1-periodic .
Proof: Suppose the solution +=t
strrtdssfeyety
0
)(
0 )()( is 1-periodic, then 0)1( yy = .
Hence,
=+
1
00
)1(
0 )( ydssfeye
srr
.
Therefore,
=1
0
)1(
0 )()1( dssfeyesrr .
Since 0r , we have 0)1( re . Hence,
= 1
0
)1(0 )(
1
1 dssfee
y srr
.
So, if )(ty is 1- periodic, then 0y must be equal to 1
0
)1( )(1
1dssfe
e
sr
r, and hence, 0y is
unique.
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Conversely, if
=
1
0
)1(
0 )(1
1dssfe
ey sr
r, we will show the
solution +=t
strrtdssfeyety
0
)(
0 )()( is 1- periodic by showing )0()1( yy = .
We have:
+=
1
0
)1(0 )()1( dssfeyey
srr
+
=
1
0
)1(1
0
)1( )()(1
1dssfedssfe
ee srsr
r
r
+
=
1
0
)1()(1
1dssfe
e
e srr
r
=
1
0
)1( )(1
1dssfe
e
sr
r
,0y=
so, we can easily to see that )(ty is 1- periodic. QED
1.2 The initial value of the periodic solution
Remark: In the general case, if )(tf isp- periodic, then the initial value of the unique
p- periodic solution is:
=p
spr
prdssfe
e
y
0
)(0 )(
1
1.
Next, we will use this result to solve problems in the real case.
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1.3 Applications
1.3(a) Fish in a lakeThe mass of fish in a lake, if left alone, increases 30% per year. However, commercial
fishing removes fish with a constant rate of 15,000 tons per year. What is the amount of
fish initially, so that there will still be fish in the lake?
Solution: We know, there is a unique initial value 0y so that the fish in the lake is 1-
periodic.
If the initial amount of fish > 0y , then the fish will grow.
If the initial amount of fish < 0y , then the fish will be gone.
What is 0y ? We have the population equation:
=
=
0)0(
000,15)(3.)('
yy
tyty
The unique initial amount is:
=
1
0
)1(0 )(1
1 dssfee
y srr
=
1
0
)1(3.
3.000,15
1
1dse
e
s
000,50= (tons)
1.3(b) Population in a village
Population of a village: 0)0( yy = .Let the birth rate = 2% , and the death rate = 1%, then
the growth rate = 1%. However, each year the number of people leaving for cities is
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ttf10
2cos30)(
= (Periodp = 10).
What is the (initial) population of the village, so that the village wont become empty?
Solution: First, we need to find what the initial value 0y is. We have the population
equation:
=
+=
0)0(
10
2cos30)(01.0)('
yy
ttyty
.
The unique initial amount is:
=
10
0
)10(01.0
)01.0(100)(
1
1dssfe
ey s
= dssee
s )10
2cos30(
1
1 10
0
)10(01.0
1.0
)])10
2cos(30[
1
1 10
0
)10(01.010
0
)10(01.0
1.0dssedse
e
ss
=
= )])10
2cos(|
01.030[
1
1 10
0
)10(01.0100
01.0
1.0dsse
e
e
ss
= dssee
ee
s )10
2cos(
1
1)1(3000
1
1 10
0
)10(01.0
1.0
1.0
1.0
= dssee
e
s )
10
2cos(
1
13000
10
0
01.01.0
1.0
(Using22
)sincos(cos
na
nunnuaenudue
auau
+
+= )
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=22
1.0
1.0)
10
2()01.0(
)1(01.0
1
13000
+
e
e
=22
)10
2()01.0(
01.03000
+
= 0253.03000 000,3
When the village has 3,000 residents, then the population of the village is not decreasing.
QED
From the above applications, we think it is important to study linear equations, not
only in 1 dimension, but in the multi-dimensional case. Before doing that we are going to
study the space nR , the nn matrices and their properties.
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CHAPTER 2: Matrices
In this chapter, we will study n-dimensional spacen
R , nn matrices and their
properties.
2.1 Space nR , nn Matrices and Their Properties
Definition 2.1: The space nR is the set of all ordered n-tuples of the form:
},,,{ 21 nuuuu L= ,
where Rui for ni 1 and Nn (the set of natural numbers). Elements innR are
called vectors.
In nR we define the dot product of two vectors ),...,,( 21 nxxxx = and
),...,,( 21 nyyyy = as follows:
nnyxyxyxyx +++= L2211 .
The norm of a vectorx in nR for each ),,( ,32,1 nxxxxx L= is define by:
222
21
|||| nxxxxxx +++== L .
The norm of a vector x in nR has a lot of properties that make it useful in
applications. In the following we collect some important properties of the norm.
Theorem2.2 (Properties of the norm) The following statements hold:
1) For each vector x in nR , 0=x if and only if 0|||| =x .
2) Ifis a real number, then |||||||||| xx = .
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3) Triangle inequality: For any two vectorsx andy in nR , we always have:
|||||||||||| yxyx ++ .
4) Schwarz Inequality: Ifx and y be vectors in nR , ),,,( 21 nxxxx L= and
),,,( 21 nyyyy L= ,
Then,
|||||||||| yxyx .
Definition2.3 Thedistance between x and y in nR is defined by :
2222
211 )()()(||||),( nn yxyxyxyxyxd +++== L
Definition2.4 (Convergence in nR ): We say that a sequence 1}{ nnx of vectors
converges to a vectorx in nR , written by xxnn
=
lim if 0||||lim =
xxnn
.
Next, I introduce the definition of norm of an nn matrixA.
Definition2.5 Let nnijaA = ][ be nn matrix,
=
nnnn
n
n
aaa
aaa
aaa
A
L
MMMM
L
L
21
22221
11211
.We considerA
as a vector in2
nR by ( )nnnnn aaaaaaA ,,,,,,,, 2111211 LLL= , (2
n terms).Then,
the norm ofA, denoted by |||| A is
==
=n
ji
ijaA
1
1
2|||| .
Theorem 2.6 LetA andB be nn matrices and x in nR , then the following inequalities
hold:
1) |||||||||||| xAAx ,
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2) |||||||||||| BAAB .Proof:
1)
The Schwarz Inequality says |||||||||| yxyx
, i.e.
( ) ( ) ( )22221222212
2211 nnnn yyyxxxyxyxyx +++++++++ LLL .
Now we have
),,,( 1122221211212111 nnnnnnnn xaxaxaxaxaxaxaxaAx ++++++++= LLLL .
According to the Schwarz Inequality, we obtain:
211
22121
21212111
2 )()()(||)(|| nnnnnnnn xaxaxaxaxaxaxaAx ++++++++++= LLLL
++++++++++++++ LLLLL ))(())(( 222
21
22
221
222
21
21
212
211 nnnn
xxxaaxxxaaa
))(( 22
21
221
nnnnn
xxxaa +++++ LL
))((22
21
221
22
221
21
212
211
nnnnnnn xxxaaaaaaa ++++++++++++= LLLLL
= .||)||||(|||||||||| 222 xAxA =
Taking the square roots, we have:
|||||||||||| xAAx .
2) Let nyyy ...,,, 21 be the column vectors of matrixB. Then it is easy to see that 1Ay
,2Ay , , nAy are the column vectors of matrix BA . Moreover, by definition we have:
==
n
iiyB
1
22 |||||||| and .||||||||1
22 ==
n
iiAyBA
On the other hand, using the above results we have:
222.|||||||||||| ii yAyA
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for i = 1, 2, , n. Hence, we obtain:
==
n
iiAyAB
1
22 ||||||||
=
n
iiyA
1
22 ||||||||
==
n
ii
yA1
22 ||||||||
22 |||||||| BA = . QED
If AB = , then we have 22 |||||||||||||||||||| AAAAAA == . With the same reasoning, we
can conclude thatnn
AA |||||||| for all natural number n.
Theorem 2.7 (Continuous Rule)
Let )(tF be a matrix-valued continuous function and )(tx be an n-dimensional
continuous function (values in nR ), then the n-dimensional function )()( txtF is
continuous.
Proof: We show that )()()()(lim axaFtxtFat=
, which is equivalent to
0||)()()()(||lim =
axaFtxtF
at
.
Then we have: ||)()()()()()()()(||lim axaFaxtFaxtFtxtFat
+
))()()(())()()((||lim aFtFaxaxtxtFat
+=
||)())()((||lim||))()()((||lim axaFtFaxtxtFatat
+
,0||)(||||)()(||||))()((||||)(|| limlim =+
axaFtFaxtxtFatat
and the theorem is proved. QED
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2.2 Derivative of n-dimensional function
Definition2.8 (Derivative of n-dimensional function)
We say nRRf : is differentiable at tifh
tfhtf
h
)()(lim
0
+
exists in nR . The limit is
called the derivative of )(tf , denoted by )(' tf .
It is easy to see that if
=
)(
)(
)()(2
1
xf
f
xf
tf
n
x
M, then
=
)('
)('
)('
)('2
1
xf
xf
xf
tf
n
M. Here is an example: If
=
tttf
cos)(
2
, then
= t
ttf
sin
2)(' . Similarly, we say )(tF is an anti-derivative of an n-
dimensional function )(tf if )()(' tftF = . Correspondingly, we define
=b
a
b
a
b
an
b
a
dttfdttfdttfdttf )(...,,)(,)(:)( 21 .
Theorem 2.9 (Product Rule)
If )(tF is a matrix-valued function and )(tx is an n-dimensional function, which are both
continuously differentiable, then )()()( txtFty = is continuously differentiable and:
)(')()()(')()( txtFtxtFtxtFdt
d+= .
Proof:h
txtFhtxhtFtxtF
dt
d
h
)()()()()()( lim
0
++=
htxtFhtxtFhtxtFhtxhtF
h)()()()()()()()(lim
0+++++=
h
txhtxtF
h
htxtFhtF
hh
))()()(()())()((limlim
00
++
++=
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h
txhtxtFhtx
h
tFhtF
hhh
)()(lim)()(lim
)()(lim
000
+++
+=
).(')()()(' txtFtxtF += QED
To reach our end result, we need to know what the eigenvalues and eigenvectors of
an nn matrixA are.
2. 3 Eigenvalues and Eigenvectors of a Matrix
Definition 2.10 (Eigenvalues and Eigenvectors of a Matrix)
Let A be an nn matrix. A scalar is called an eigenvalue ofA if there exists a nonzero
vector x such that
xAx = .
The nonzero vector x is called an eigenvectorcorresponding to . We also note that any
scalar multiple of the eigenvector is also an eigenvector. Finding eigenvalues can be
simplified into a general process as follows: From xAx = , we have 0)( = xA
for 0x . Therefore, A is a singular matrix, or equivalently:
.0)det( = A
This equation is called the characteristic equation. Solutions to this equation will be
eigenvalues.
For each eigenvalue, there is one or more corresponding eigenvectors (we disregard the
multiplicity). Here is an example: Find the eigenvalues and corresponding eigenvectors
of the matrix
=
32
41A . Then,
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).1)(5(
548)3)(1(32
41
32
41
0
0|A| 2
+=
==
=
=
I
This gives two eigenvalues 15 21 == and . Then, we need to find the corresponding
eigenvectors. That means we need to solve the homogeneous linear
system 0) = xAI for each eigenvalue.
For 51 = we have
=
=
=
0
0
22
44)
32
41
50
05()(5
2
1
2
1
x
x
x
xxAI .
Solving that system, we get
=
1
1cx .
For ,12 = we have
=
=
=
0
0
42
42)
32
41
10
01()
2
1
2
1
x
x
x
xxAI .
Solving that system, we get
=
1
2cx .
Next, we will study the matrix exponential function tAe .
2.4 Matrix Exponential FunctiontA
e
LetA be a nn matrix. What are the matrixA
e and the functiontA
e ? We have
different approaches to define these matrices.
Definition 2.11 Suppose A is an nn matrix. Then the matrix Ae is defined by:
=++++=
= 0
2
!)
!!2!1(lim
n
nn
n
A
n
A
n
AAAIe L
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and
=
=0 !n
nn
tA tn
Ae .
The above definition is meaningful. Indeed, if we denote!!2!1
:2
n
AAAIS
n
n ++++= L ,
then
=
+= 1 !ni
i
n
A
i
ASe and hence,
||!
||||||1
=
+=ni
i
n
A
i
ASe 0
!
||||
!
||||
1 1
+=
+=ni ni
ii
i
A
i
Aas n .
Using the above definition we will find tAe for some given matrixA.
Examples 2.12: Find tAe , if a)
=
01
10A
b)
=
01
10A
c)
= 11
11A
a) If
=
01
10A then
=
01
102A ,
=
10
013A and IA =
=
10
014 .
From that pattern we have AA =5 , 26 AA = ,., and so on. Hence we obtain
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.cossin
sincos
)!2()1(
)!12()1(
)!12(
)1(
)!2(
)1(
10
01
!401
10
!310
01
!201
10
10
01
0
2
0
120
12
0
2
432
=
+
+
=
+
+
+
+
+
=
=
=
+
=
+
=
tt
tt
n
t
n
tn
t
n
t
tttte
n
nn
n
nn
n
nn
n
nn
tAL
Here we have used the facts that =
=0
2
)!2()1(cos
n
nn
n
tt ) and
+=
=
+
0
12
)!12()1(sin
n
nn
n
tt .
b) If
=
01
10A then L,,,
10
01,
01
10432 IAAAIAA ===
=
= . Hence,
...10
01
!401
10
!310
01
!201
10
10
01 432+
+
+
+
+
=
tttte tA
=
tt
tt
coshsinh
sinhcosh.
(Using .2
cosh;2
sinhtttt
eet
eet
+=
= ) .
c) If
=
11
11A then
=
=
00
00
11
11
11
112A ,
=
00
003A ,
=
00
004A , L .
Hence,
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.1
1
10
01
00
00
!400
00
!300
00
!211
11
10
01 432
+=
+
=
+
+
+
+
+
=
tt
tt
tt
tt
tttte tA L
.
a) Finally, if
=
00
10
01
100
0010
LLL
OLM
OOM
MO
L
A , then
.
100
1)!2(
10
)!1(!21
2
12
=
LL
OOOM
MOOM
L
L
t
n
tt
n
ttt
e
n
n
At
Theorem 2.13 (Properties of the matrix exponential function)
LetA andB be nn matrices and s and tbe real numbers. Then
(a) IeO = (O is the zero nn matrix);(b) Iee ttI = ;(c) AsAtstA eee =+ )( ;(d) .)( 1 AtAt ee =
Proof: (a) Using the formula ...!
)(
!2
)(
!1
2
+++++=n
tAtAtAIe
ntA
L , we can calculate
.....!
...!2!1
2
+++++=n
OOOIe
nO
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=I.
(b) From .....!
)(...
!2
)(
!1
2
+
++
+
+=n
AtAtAtIe
ntA we have
.....!
)(...
!2
)(
!1
2
+++++=n
IrIrIrIe
nrI
= .....!
...!2!1
2
++++
+n
IrIrIrI
n
= .....)!
...!2!1
1(2
+++++n
rrrI
n
= Ie r . QED
(c) 1) If A is diagonal, i.e.
=
n
A
00
0
0
00
2
1
L
OOM
MO
L
. Then,
=
tn
t
t
At
e
e
e
e
00
0
0
00
2
1
L
OOM
MO
L
and
=
sn
s
s
As
e
e
e
e
00
0
0
00
2
1
L
OOM
MO
L
.
Hence, we can obtain
=
+
+
+
)(
)(2
)(1
00
0
0
00
stn
st
st
AsAt
e
e
e
ee
L
OOM
MO
L
= )( stAe + .
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IfA is diagonalizable, then there exists S, 1S , and ISS = 1 and DSAS =1 , a
diagonal matrix. Then we have DSSA 1= . We show now that 212 DSSA = ,
313
DSSA =
, ,nn
DSSA =
1
. Indeed, we have
22111112 )( DSASSASSASSAASSSA ==== .
Using the same reasoning, we can conclude that that nnn DSASSSA == )( 11 , a diagonal
matrix. Hence,
tD
n
nn
n
nnn
n
nAt en
Dt
n
StSAS
n
tASSSe ====
=
=
=
00
11
0
1
!!)
!( .
Thus, we have SeSetDAt 1= , SeSe sDAs 1= and SeSe stDstA )(1)( ++ = . Therefore,
SeSSeSee sDtDAsAt11 =
= SeeS sDtD )(1
= SeS stD )(1 + . (sinceD is a diagonal matrix)
)( stAe += .
2) LetA be a Jordan block, then
+
=
=
00
10
01
100
0010
00
0
0
00
000
1
000
10
001
LLL
OLM
OOM
MO
L
LL
OOOM
MOOOM
MOO
LL
L
OOMM
O
MO
L
A ,
BI+= . .
. We observe that it is not hard to show is any constant, then)( AIAI
eee+= . So we
can obtain
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22
===
+
100
1)!2(
10
)!1(!21
2
12
)(
LL
OOOM
MOOM
L
L
t
n
tt
n
ttt
eeeee
n
n
IttBItBItAt
and similarly,
===
+
100
1)!2(
10
)!1(!21
2
12
)(
LL
OOOM
MOOM
L
L
s
n
ss
n
sss
eeeee
n
n
IssBIsBIsAs .
Hence, we obtain
=
100
1)!2(
10
)!1(!21
100
1)!2(
10
)!1(!21
2
12
2
12
LL
OOOM
MOOM
L
L
LL
OOOM
MOOM
L
L
s
n
ss
n
sss
e
t
n
tt
n
ttt
eee
n
n
Is
n
n
ItAsAt
,
=
100
1
)!2(10
)!1(!21
100
1
)!2(10
)!1(!21
2
12
2
12
LL
OOOMMOOM
L
L
LL
OOOMMOOM
L
L
s
n
ss
n
sss
t
n
tt
n
ttt
ee
n
n
n
n
IsIt ,
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( )
+
++
+++
=
+
100
)(1
)!2(
)()(10
)!1(
)(
!2
)()(1
2
12
LL
OOOMMOOM
L
L
st
n
stst
n
ststst
e
n
n
Ist ,
AstBstIsteee
)()()( +++ == .
Finally, ifA is similar to the Jordan block, i.e. 1= SJSA with J=Jordan block. Then we
have,
AsAtJsJtJsJtJsJtstJstA eeSSeSSeSeeSSSeSSee ===== +++ 11111)()( )( .
Therefore, the proof is completed.
c) We already proved AsAtstA eee =+ )( . Let ts = , we have
IeeeeOttAtAtA === )(. .
Hence, AtAt ee =1)( . QED
From property
AsAtstA eee =+ )( we have22 )( tAtAtAtA eeee == . Similarly, ntAntA ee )(= for
any natural number n. This formula will be used often later.
Theorem2.14 IfA is a nn matrix, then tAtA Aeedt
d= .
Proof: We know LL +++++=!!2!1
12
n
AAAe
nA . Then we have
LL +++++=!
)(
!2
)(
!11
2
n
tAtAtAe
ntA
Since the above Taylor series converges, and the series of derivatives of these terms
converges too, we have:
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( ) ( )( )
.
)!1!21
1(
)!1(121
1)1(123
3
12
2
10
12
1322
1322
tA
n
nn
nntA
Ae
n
tAtAtAA
n
AtAttAA
nn
AntAttAAe
dt
d
=
+
++++=
+
++
++=
+
+
+
++=
LL
LL
LL
L
QED
2.5 The Cauchy Integral Formula of Exponential Function
The second approach to define tAe is using the Riesz theory. Recall, the Cauchy
Integral Formula is a useful tool for solving many problems in Complex Analysis. The
Cauchy Integral Formula formally states that given a complex function )(zf that is
analytic everywhere inside and on a simple closed contour C, taken in the positive sense,
with 0z being interior to C, the following (the Cauchy Formula) is true:
dzzz
zf
izf
c
=
00
)(
2
1)(
.
The Cauchy Integral Formula states that the value of )( 0zf can be determined, if )(zf on
a closed contour around 0z is known. An additional formula, the Cauchy Integral
Formula for derivatives, is given below.
( ).
)(
2
1)('
2
0
0 dzzz
zf
izf
c
=
In order to accommodate later use, the Cauchy Integral Formula can be rewritten as
( ) .)(2
1)(
1
00 dzzzzfi
zfc
=
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Now, ifA is a nn matrix, and )(zf is an analytic function in a domain containing
eigenvalues of A., we define the matrix-valued function )(Af by:
( )dzAzzf
iAf
c
= 1)(2
1)(
,
where Cis a closed contour containing the eigenvalues of A. By the above definition we
have:
.)(2
1 1 =
C
zAdzAze
ie
Note that the definition is independent of the choice of the contour C. We will find out
that the two above definitions of Ae using the two approaches are the same. First, we
study some properties of )(Af .
Theorem2.15 The following statements hold:
1. If 1)( =zf , then IAf =)( .2. If zzf =)( , then AAf =)( .3. If nzzf =)( , then nAAf =)( .Proof:
1. We know. if 1|||| for allz on C. We have then 1=++< ayxyx .
Corollary4.6 IfX is a Hilbert space, then
(a) 0|||| =x implies 0=x .
(b) |||||||||| xx = for in Fand x in X .
(c) (Triangle Inequality) |||||||||||| yxyx ++ for yx , in X ,
Next we define linear operators on Hilbert spaces.
Definition4.7 An operatorA from a Hilbert spaceHto another Hilbert space Kis called
linear if it satisfies the following conditions:
a) ;)( AyAxyxA +=+
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b) AxxA =)( For allx,y inHand in F.
Definition4.8 A linear operatorA is said to be continuous, if xxn inHimplies
AxAxn .
By using the standard argument we can prove this theorem.
Proposition 4.9 ([2], Proposition II.1.1) Let Hand Kbe Hilbert spaces andA:
KH a linear operator. The following statements are equivalent.
(a)A is continuous.
(b)A is continuous at 0.
(c)A is continuous at some point.
(d) There is a constant 0>c such that |||||||| hcAh for all h in H. We say in this case
thatA is a bounded operator.
Definition4.10 (The norm of a bounded operator) LetA be a bounded operator. The
norm ofA, denoted by |||| A is defined by:
}1||||,||:sup{|||||| = hHhAhA .
Remark: It is not hard to see that
}1||||||:sup{|||||| == hAA
}0||:||/||sup{|| = hhAh
}.||,||||:||0inf{ HinhhcAhc >=
The following theorem is about properties of the norm of bounded operators. First, the set
of all bounded operators fromHto Kis denoted by ),( KHB . If HK= , then we denote
)(HB the set of all bounded operators fromHto itself.
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Proposition 4.11 ([2], Proposition II.1.2)
(a) IfA andB ),( KHB , then ),( KHBBA + , and |||||||||||| BABA ++ .
(b) If F and ),( KHBA , then ),( KHBA and |||||||||| AA = .
(c) If ),( KHBA and ),( LKBB , then ),( LHBBA and |||||||||||| ABBA .
We also can define a vector-valued function HRtf :)( . The continuity and the
derivative of such functions are defined as in nR . Also, the continuity and the product
rule in Theorem 2.7 and in Theorem 2.9 also hold in Hilbert space.
2. The Spectrum of an Operator
Definition 4.12 Let H be a Hilbert space and )(HBA . The resolvent ofA, denoted
by )(A , is the set of all complex numbers such that )( AI has an inverse and
1)( AI is also a bounded operator.
The complement set of the resolvent in Cis called the spectrum of A, denoted by )(A ;
)(\)( ACA = .
An operatorA is called injective if AyAx for yx , and called surjective if the range
ofA is the whole spaceH. It is not hard to see that is in resolvent set if and only if
)( AI is both injective and surjective.
Let now denote the kernel space ofA.
}0:{)ker(==
AxHxA .
It is easy to see that )ker(0 A for every operatorA. If }0{)ker( =A , thenA is injective.
Definition 4.13 The point spectrum ofA , )(Ap , is defined by
{ })0()ker(:)( = ACAp
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As in the case of operators on a Hilbert space, elements of )(Ap are called eigenvalues.
If )(Ap , non-zero vectors in )ker( A are called eigenvectors; )ker( A is
called the eigenspace ofA at .
It is well known that. if 1|||| , then we have 1