Qualitative Behavior of Solutions to Differential Equations in -i

8/6/2019 Qualitative Behavior of Solutions to Differential Equations in -i

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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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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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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



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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2

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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16

).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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19

.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 + .


27/66

21

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


28/66

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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23

( )

+

++

+++

=

+

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:


30/66

24

( ) ( )( )

.

)!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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25

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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48

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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49

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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50

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

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