Methods of Matrix Algebra

Methods o f MATRIX ALGEBRA Stanford Research Institute Menlo Park, California

1965

York

0 1965,

111

10003

United Kingdom Edition published by

W.l OF

65-19017

Foreword book, on

go

book

by

up

why up

FOREWORD

no up book

book, ;

Also,

on on on

on

on

by no

by

by ;

...

on

on on

u

=

Tv

T

by

book on

book

by

FOREWORD

by on

of

by upon do by body on

on on

book on

of

book from

through

FOREWORD

no go

1

book

PEASE

Symbols and Conventions A X

x*

A*

xT

AT

xt

At

x

x x

A.

A.

A.

p.

0

...”

E

xi

X -i

4, pp. pp. 49, 217.

A

B, (AB - BA), p. 279. [A,

xi

p. 305.

CHAPTER I

Vectors and Matrices by of

1. VECTORS

As

by

Definition.

A vector is a set of n numbers arranged in a de$nite

order. n

F,” components n

dimensionality

by

scalars.

1

2

I. VECTORS A N D MATRICES

F by on x = col(x, x2 ... xn)

(1’)

T transpose

As components,

box 1.

box

1.

3

VECTORS

El FIG. 1.

x1

A 2-port network.

x2 =

=

(?)

(3)

by no

by

by

e

=

(Ej,

i

=

e

w,

E’s

1’s

E,

#1 do go

#1

E, 1,

.

4

AND

on

x1 of the I

#1 by n,

N, (N by

(N -

n

vector

by

+') by

on up

1. on,

2.

5

on of

2

?

3 do

by

2. A D D I T I O N O F VECTORS A N D SCALAR MULTIPLICATION

Definition. The sum of two n-dimensional vectors x and y is the vector whose components are the sums of the components of x and y . That is, if

then x1 +Y1

(7)

F,

6

I . VECTORS AND MATRICES

no Definition. The product of a scalar a, in a field F , and a vector x, whose components are in F, is the vector whose components are a times the components of x. If

x =

(8)

then

a

F,

xi

1.

commutative

associative:

x+y=y+x x+(Y+z)=(x+y)+z

2.

S, F x

null vector 0,

S,, x+O=x

O=

3.

x

(I)

S,

negative, x

+ (-x)

=0

(-x)

7

3.

x

by

4.

associative:

(13)

a(bx) = (ab)x

5.

distributive (a

+ b)x = ax + bx

a(x

+ y)

= ax

+ ay

F,

6. l.x=x

7.

1, (15)

commutative:

ax

go

= xa

(16)

on go

per se,

do by 3. LINEAR VECTOR SPACES

Definition. A set S of n-dimensional vectors is a linear vector space over the field F if the sum of any two vectors in S is in the set and if the product of any vector in S times a scalar in F is in S .

8

I.

S,

y

x

p

a

F,

(ax

+ fly)

S.

F,

F, , F2. (ax

+ by)

a

b book,

p,

p

characteristic p ) .

p binary field, do

F F book,

4. DIMENSIONALITY A N D BASES

not

=

2,

4.

9

DIMENSIONALITY AND BASES

Definition. The set of k n-dimensional vectors x1 , xz , ..., xk are said to be linearly independent in the jield F if and only if there exists no set of scalars c1 , c2 , ..., ck of F , not all zero, such that ClX1

+

CZXZ

+ + "'

C/:X/c =

0

(17)

ci ci

S. Definition. A linear vector space S has the dimension k if S contains at least one set of k vectors, none of which are the null vector, that are linearly independent, but does not contain any set of I ) linearly independent vectors.

+

Definition. If S is R-dimensional, then a set of vectors in S , x1 , x2 , ..., xk that are linearly independent is called a basis f o r

S generated by linear envelope span

over thejield

k

k basis

y y

= CIXl

+ c2x2+ ." +

no

ci

y

xi

y

ci

xi, A

(18)

CkXk

xi xi

complete

10

I.

xi

y expansion

xi.

y on

n

k

n

k.

0 < k < n,

the whole space. proper subspace no

ux

subspace.

null space,

+ by + cz = 0

(19)

on

on linear mani-

fold

no on

5. L I N E A R H O M O G E N E O U S SYSTEMS-MATRICES

do

5.

11

LINEAR HOMOGENEOUS SYSTEMS-MATRICES

1.

1

x2, x2 .

by

x1

x2

x2.

on

do

As

box

1,

no

(21) by

x1

linear

x2

a,

f(ax2) = ax,

by

good

12

I . VECTORS A N D MATRICES

?

homogeneity.

x, #1

x,

x,

x1 1

x1 El I,

+ BIZ = CE, + nr,

=

AE,

A, B, E,

E,

I,.

I,. x, by maps

x, .

mapped onto

x, maps S , into

, Eq.

x,

onto S , .

.

x, S,

,

5.

y1

= %lJl

y2

=

E D,

4-a12x2 t

a,,%,

ym = %lXl

A, B,

13

LINEAR HOMOGENEOUS SYSTEMS-MATRICES

+ a2pxz 4 +

arn2X2

"'

"'

t n,,,x,1 a2pn

+ ... +

GITlxn

E a,, , do

?

do

E,

I,,

xl,

..., x, ,

any any do

any

matrix A

=

A

(c

B

D)

= (aii)

(26)

14

I . VECTORS A N D MATRICES

on

main diagonal

A.

aii . on

diagonal. on

by A

, u P 2 ,..., unn)

=

(27)

on

dimensionality

by

m x n

A A on

n

(x,

, ..., xn)

“ m by n”)

, ..., y J .

m

A A 2 x 2 A

B

A

B

A , B

...

on

by

2 x 2

x2

on

S,

x1

S, I

x1

x,

.

6.

15

PARTITIONED MATRICES

by

6. PARTITIONED MATRICES

on up

A

=

N

&I m x m

A

(m

M R N)

(s

+ n) x ( m + n).

R,

n x n m x n,

S n x m.

no

partitioned

A

A

A, A

quasidiagonal.

(29)

on

A

= quasidiag(A, , A ? ,

-*a,

AP)

V on

so

by no

(30)

16

I. VECTORS AND MATRICES

m x n m x 1 A

=

(xi xz

9

. *)

(31)

xn)

A,

xi on

A

A =

i"i YmT

on

A

7. ADDITION O F MATRICES A N D SCALAR MULTIPLICATION

Definition. The sum of A = ( a i j )and B = (bij)of the same dimensions is the matrix whose terms are the term-by-term sum of the terms of the separate matrices:

(33) Definition.

The product of A = ( a i j ) times a scalar a is the matrix

aA = ( a a i j ) : (34)

17

8.

do

by 8. MULTIPLICATION O F A MATRIX T I M E S A VECTOR

(24) (22) ?

1 maps

x2’sonto

xl’s.

xl”

x2

(22) (24)

operating on x2

y = 2x,

y

by

by

x

x,

xl,

y’s

x’s

by x1 = NIX,

(35)

x2. ?

y

=

AX

y

A by

by

x

by

18

VECTORS

MATRICES

x. y. no

by AX)

=

(orA)~= A((Yx)= AX)^

(40)

no

by

3’2

“.

= (XI

x2

XI?) x

“.

i:“ ::j n,,

yT

==

1:

a2n

’.’

= %,a,,

amn

(41)

xTAT

yT xT by yz

I;

+

x2a2.2

+ ”. +

by

by

w 4 n

(42)

(41) by on a,, 1s

, aZ2,

aji .

transposes

A

on m x n

n x m

n

m m

n

uij ,

(41)

8.

19

MATRIX MULTIPLICATION

x 1 x n

by

xT

n x 1

Eq. Eq. (37) Eq. (41)

AX)^ = x’A”

(43)

9. MATRIX MULTIPLICATION

by on on z.

y,

,to z

x.

x.

z product

y

=

AX

z

=

By

x.

20

I . VECTORS A N D MATRICES

p A B. 1

I, up

A

2, B.

of y

(47) (4), R R

(45)

2 by

(47),

x.

z z

(45)

=

By

=

B(Ax) = (BA)x

(47)

(48)

9.

21

MATRIX MULTIPLICATION

(BA),, B

A (BA)ij = Z b i k a k j

(52)

I,

BA B by

by

A

C

=

2 3 6 7 (4 5 ) ' (8 9)

BA 2.6+3.8

x

2.7+3.9 4.7+5*9

(4 . 6 + 5 * 8

=

=

n

(54)

Ii:

14 + 2 7 28 45)

+

xp

n x n

1.

2.

3.

+ B)C = AC + BC A(B + C) = AB + AC (A

A(BC) = (AB)C

=

36 41 (64 73)

nxp

22

I. VECTORS A N D MATRICES

4.

null matrix,

A, Q * A =A . Q = Q

5.

identity matrix,

A,

IA

=

A1

=

A

I

Eq. on 1 0 0 0 0 1 0 0

... =

(S,J

(55)

Sii Sj,

=

= 1

0

i#j .

z = j

.

Also,

10. 4N ALGEBRA

n x n algebra product relation, has

of * A ring

(b)

no

1 1.

23

COMMUTATIVITY

n x n

n x n

by

A

n2

n x n

B F n x n

cy

/3

(cuA + PB)

F,

F, of

A

B

AB.

associative algebra.

x(yz) # (xy)~.

n x n

of

1

Eij

A A

A

=

=

(aij)

r)~ijEij ii

on

Eij no

cu

C C ~ , ,=E0~ ~ ij

11. COMM UTATlVlTY

F

;i n x n (57)

24

I . VECTORS AND MATRICES

AB # BA. on

As

6.2+7.4 6.3+7.5 8.219.4 8.3+9.5

1

=

21.

40 53 (52 69)

never

do n x n commutative subspace

n x n n x n on

2 of

up no

As

12.

25

OF

A(B + C)

AB

AC,

(A + B).

As

(A

+ B)' = (A + B)(A + B) = A' + BA + AB + B2

(59)

no

BA

AB a

12. DIVISORS O F Z E R O

by 1 0 0 0 0 0 (0 O N 1 0 ) = (0 0)

by

nilpotent.

X). A

Ax

not

A

= 0,

x Ax

=

Axi

=0

0

any

xf

A

i,

A

26

I.

13. A M A T R I X AS A R E P R E S E N T A T I O N O F A N ABSTRACT OPERATOR

4

on

#I

1,

#2. abstract operator.

by arepresentation of the abstract operator. 1

El ABCD matrix

I, ,

E,

I,

E's

transmission matrix,

Eq. (4),

E-I basis.

on

I's

wave matrix

on

wave basis.

E

+ RI R.

E

-

RI

3,

"-i

-u1

FIG. 3.

s

Waves at the terminals

-:-I a 2-port network.

14.

Eq. on

27

OTHER PRODUCT RELATIONS

scattering matrix scattering basis.

E’s I’s. impedance matrix. admittance

E2,

representations ABCD

El

I,

E,

I1

I,

I, , El

abstract operator.

V. 14. O T H E R P R O D U C T R E L A T I O N S

2.

4.

28

AND

-

-

#2

I 1 J

S

(61)

on

on

S (61)

by

A,

v2

u2

A, A1

S S,

(62),

=

aid,

-

blcl ,

A,

=

a,d, - b,c,

(65)

(62),

star product of

S, ,

s

=

s, * s,

(66)

9

15.

29

T H E INVERSE OF A MATRIX

on

A

15. T H E I N V E R S E O F A M A T R I X

1. #2,

x1

x2 A:

x2

x1 =

x2 on xl. x2 =

A-l

inverse

A.

(68), x2 =

x1 =

=

=

=

=

x2, = M-1 =

I

1

(55). A-'

E,

I, y l , ...,y,

El

I,, m

n

n

x,

, ..., x,, ,

=

n. xl,

..., x, .

30

I . VECTORS A N D MATRICES

y l , ...,yn x l , ..., x , i f and

(23) only ;f a11

(A\=

a12

a1n

"'

: Qn1

: #O an2

'.'

(72)

ann

A.

Aij

A-l, by

+1

i

+j

by

ij

A-1

=

by A I. A-'

-1

i

+j

odd, by

(74)

Definition. A square matrix A is singular i f its determinant is zero. I t is nonsingular if its determinant is not zero.

Theorem. Given a nonsingular square matrix A, its inverse exists uniquely. Furthermore, its left inverse is the same as its right.

on on

16.

I aij I

31

RANK OF A MATRIX

I bij I

by

(54) A

B IABl = / A l . I B / by

(54).

(76), AB

A

B

AB.

Theorem. If A, B, C, etc., are n x n square matrices, all of which is the product are nonsingular, then the inverse of the product (ABC of the inverses in reverse order. a * * )

X

=

(ABC ...)X by A-l,

X

(77)

(ABC ...)-' =

I

by B-l,

by C-l,

... C-lB-lA-1

~.

(79)

16. RANK O F A MATRIX

n x n

A.

rank k x k,

A (n - k )

k

no

A

32

I . VECTORS A N D MATRICES

k x k

so

by

n x n

A

A

n. (n - k)

n,

17. GAUSS’S ALGORITHM3

15

A by

A,

Ax x,

(80)

=y

y.

Eq.

a,, = 0, x1 # 0.

a,, # 0 x1

(i # l ) ,

x,

(ail/all)

b’s

so y.‘

, z

= y2

-

yi’

by

a tlYl/all .

S A n algorithm is a procedure whereby a desired result can be obtained in a finite usually restricted, however, to procedures number of computable steps. The term that are practical for at least computer calculation.

17.

33

GAUSS’S ALGORITHM

b,,

x,

x’s

x2

b’s

b,, # 0.

x,

allXl

+ + hz,x, + hzsx, + ... + bznxn + ... +

i -n l $ 2

a13x3

”‘

1alrrXn = y1

c3nxn

c33x3

= yz’

,,

= y3

i“!;j !j(i)011

a12

a13

:::...

-

...

..

Xn

fnn

on on

A upper triangular.

x,

(83)

y.

x n = y F - 1 )if,n

do x,

fnn

# 0. xnP1,

on, u p

34

I . VECTORS AND MATRICES

do

on

y. A-l. y1

xi

AX^ i

1

n.

=

(84)

y+

A-l

xi. up

on do,

x

(85),

y. y; = y&”= 0.

do

A

on

A x

book.

y

18.

35

2-PORT NETWORKS

18. 2-PORT N E T W O R K S

by

5.

FIG.5.

Partition

a ladder network.

up

2. a

Ei

E, Ii = I , =

+ RI,

36

I. VECTORS A N D MATRICES

TABLE I

TRANSMISSION MATRIX OF BASIC2-PORT ELEMENTS ON Element

Matjx

Series impedance

i:,4)

Shunt admittance

Transmission line

cos p

i

jZsin p

-cp-

(isin p

cos

Characteristic impedance Z Electrical length p

cosh

Waveguide below cutoff Characteristic impedance jX Electrical length jr

r

-sinh

Transformer

M

1.

by

jX sinh

r

cosh

r

r

i

19.

37

EXAMPLE

2. on A B E, ( C D)( I ,

El

=

I1 =

+

AE, BIZ CE, $- DI, -

E,

=

1,

=

BC)

DE, - BI, -CE, AI,

+

I,

I,

1):

A

=

D.

3. A

I.

AB. (+{El*

+ E*I})

19. EXAMPLE

As 2,

B,

38 by

2,

=

2,.

by

Y,

Z,,

2, , 2, ,

Y

Exercises

1.

8,

2, ( c ) coI(1,

I,

3,

0,

2, 4, - I ,

2.

col(0, - 1 , 0,

1,

E, ?

on on x1 , x2 , ..., x, x1

+ + ..' + x,

=0

XI

+ x, + ... + x,

=

x2

1

,

39

EXERCISES

3. of

x

= C O ~ ( U 6, , C, a,

6, C, a, ...)

a, b, c, a, 6 , c

4.

S, ,

S,

sum

S,

S,

S,

S,,

S,

union

S, , S,

S, S,

intersection

.

S, , S, .

S, ?

5. A =

AB

I B I.

(';'

B = (1 - J

'j.), 1 -J

I AB I

BA.

'

-j

I BA I

IAI

6.

7. A = ( O0 1

8. A

=

(;

by

1)

1)

1 0 A=f

a

0 0

40

I . VECTORS AND MATRICES

9.

of 1

A=(:

10. 11.

1 -1 2 -3

-1

0 -1

: ; 1;)

Eq. (64)

Ya L

==C by

12.

a A=(-b

b a)

A (AB).

a

(a

of

+ jb).

B b

by semigroup have the group property.

of

a a

group,

a = 1, b

=

0.)

13.

A=(

AB # BA

-c

a f j b jd

+

a

-

jb

b,

b

41

EXERCISES

14.

An

A

by

Bn,

B n

n

(AB)" 15.

A

d

a

Pn

16.

= ad - bc

(n

+

17.

D

n.

x, dldx. x, x=,..., xn.

D

A, B, C,

n x n

(A

X

D on

+ B) Y

AX+BY=C BXtAY = D

X 2n x 2n

18.

A,

C,

M-1

D

=

Y. M,

n x n

(A - BD-'C)-' (B - AC-'D)-'

of

(D - CA-lB):'

(A -

42 19.

M

2n x 2n

n x n

D)

=

D

(Hint:

D

by

X

C

C H A P T E R II

The Inner Product no a priori

(3) 2 3

?

do

by topological space).

1. U N I T A R Y I N N E R P R O D U C T

(9

x =

xi7

x

3

no

A

by to

43

44

11. THE INNER PRODUCT

by 11 x 11

x”, no a

11 x 11

I xi 1

xi

xi

< E,

< E. by

by a.

a,

on

As

x

y

=

y,

by

x,

!I x (1 (3)

=

(x, X)l’Z

inner product

(4)

x

y.

Many authors use X x,Y,*. We prefer the form given above since it somewhat simplifies symbology that we shall introduce in the next section. The distinction is trivial, however, providing one is careful to note which form is being used.

1.

45

UNITARY I N N E R PRODUCT

()

Eq.

x x y

x

y, /3,

FIG. 1 .

(x,y)

a

y, #1

1.

x1 = x cos a,

y1 = y cos j3

x2 = x sin a:

y3 =y

(5 1

j3

The unitary inner product of two real vectors.

+ x2y2 = xy(cos

= xlyl

01

cos j3

+

01

j3)

= xy cos(O1 - j3)

(6)

(x, x on y,

y*,

y on x, y. on

x*.

x

by

unitary inner product relation unitary space U-space

v,

x

y by

Eq.

U,

.

46

11. THE INNER PRODUCT

(x,yj

cp

x

y

cp

As

Eq. (3). (x,x> 2 0

(9)

x

Also

<x, Y> =

(Y, x>*

(10)

(x, ay> = 4x9 Y>

u

x

x

u

v, (x,u

(x, au

+ v> = (x,u> + <x,v>

+ bv)

=

a(x, u)

+ b(x, v)

by

(1 1)

linear in the second

factor.

( 1 1)

(au

+ bv, x) = a*(u, x) + b*(v, by

x) antilinear

47

1.

scalar product inner product. CauchySchwartz inequality: Given the inner product relation of Eq. ( 3 ) , then for any

Theorem.

x and y




y

x

f 0

# 0,

(ax

by

+ by):

+

(ax

+

=

+

+ +

a

b

30

by b =

a =

b*

=

=

(1 5 )

+

30

-

# 0,

2) Triangle inequality,

Theorem.

Given the inner product relation of Eq. ( 3 ) , then for any

x and y

+

+ (13)

+

+

=

<

x)l’z

+ (Y. Y)1’2}2

(17)

I pl

pl‘

2. ALTERNATIVE REPRESENTATION O F U NITARY INNER PRODUCT

XT

=

(xl xg ... xn)

(18)

T Eq. (3) (x, y)

(19)

xi*yi = x*Ty

=

(t)

xt

= X*T =

(xl*

(‘11

‘12

‘13

= aZ1

a,,

...

u31

At

hermitian conjugate

...

(20)

x,*)

hermitian conjugate

xt

A

...

x2*

x.

’.’) (21)

A.

Eq. (3) (X,Y>

= xty

(23)

3.

GENERAL (PROPER)

49

INNER PRODUCT

on

xty,

3. G E N E R A L (PROPER) I N N E R P R O D U C T

(x, x>

(A)

(B)

(x, x) (x, x)

x =

0

x

=

0

(x, y)

y

(x, Y>

=

(x, ay

+ pz> = a(x, y) + p(x, z )

x,

(Y, x > *

44).

I). (3)

K I, xtKy

K

=

kij ,

(ytKx)*

8). (25)

50

11. THE INNER PRODUCT

on

i,j

x

y,

kij

K

=

=

k,*,

Kt

A square matrix K such that it equals its hermitian conjugate Kt) is called hermitian.

Definition.

(K

=

K

(x, x)

x. good

(A) by Definition. A square hermitian matrix K is positive definite if, for all x except the null vector,

xtKx > 0

(30)

It is positive semidefinite or nonnegative definite except the null vector,

xtKx

if, for all x

0

negative definite

(31)

negative semidefinite K all x,

definite. indefinite.

nonpositive

x (of

(24) K no

K

(24)

4.

51

EUCLIDEAN I N N E R PRODUCT

K proper

(23),

K K

by

=

a

I, metric on

on on

(9),

(1

Hilbert space.

s(x) = xtsx

=

z

xi*xjsij

(32)

i3

quadrat,icform

xi

. (St = S ) xi,

S s(x) hermitian quadratic form.

4. E U C L I D E A N I N N E R P R O D U C T

(3) <x,Y>

=

z i

xiyi

Y'X

z=

(33)

52

11. T H E INNER PRODUCT

En.

a

(24) (x, y)

(34)

= x%y

S ST = S ,

S

=

sii

(sii),

=

sii

.

S

x

(A’)

y

(34)

S,

x.

(x, x)

(x,x) (x, (D’) (x,

=

=

x

0

+ bz)

x) =

a<x,

=

0.

+ b<x,

2).

(D’)

(3) (24)

(33) (A)-(D),

As

(33), x, y,

(34)

(3) (34)

(24)

(A’)-(D’)

5. a

53

SKEW AXES

b

a

b. of Eq. (33)

Eqs. (3)

(33)

x+y = 2(a,a,

+ bib,)

x’y = 2(a,a,

-

b,b,)

K xTy,

K.) do

5. SKEW A X E S

K

x

a

2. on

x1

x2, a

6, a = XI

b

+ x2 cos y

= x,siny

FIG. 2. Skew axes in two-dimensional euclidean space.

54

PRODUCT

11.

x, 1 I"

by =

a2

+ b2 = xI2 + 2x,x,

cos 'p

+ xi2

c0s7

K = (cosy

1

K.

K

by

on by

by do. no

a

(3) no a priori no

priori.

do odd on

As

(3),

E-I

5.

55

SKEW AXES

by rotational

1

K = -1( 0 1 2 1 0

(37)

K

f, . by by fn = fo

parametric harmonics

fo ,

+ nfp ,

n

=

on

0, fn

... .

. by

En

In fn

e = col(... E-,

i

P

=

=

.

by

, E, , El , ...)

col( ...I-, , I , , 1, , ...)

diag( ... l/f-l, l/f,,

...)

56

11.

THE INNER PRODUCT

x

=

=

O P (P 0 )

(43)

by z,

s

=x

O P e t ~= x (et, it)(P O i

)( )

(44)

K,

by

6. ORTHOGONALITY

by Definition. Given an inner product relation defined by the metric K, two vectors x and y are orthogonal, or precisely, K-orthogonal, when

(x, y)

=

xtKy

=

0 q~ =

0,

q~ =

Definition. Given an inner product relation defined by the metric K, the set of vectors uiis said to be orthogonal,or better, K-orthogonal, when

(uiuj)

=

u,tKuj = 0

so that each pair of vectors is K-orthogonal.

i #j

(45)

6.

57

ORTHOGONALITY

n

K

ui . by

x

x

==

z

aiuj

I

x

n

n

ai .

n ai .

ui Eq. (46) n

K

ui

k

Eq. (46) by u,tK, n

1,

..., n

ul,.tKx = z a i u l . t K u j

(47)

I

(46).

i

on a, =

Eq. (46)

(u,.tKx)/(u,tKu,)

=

K. (48)

58

11. T H E INNER PRODUCT

(46)

7. NORMALIZATION

ui x

by ui by vi ,

by

a!i :

v,

v,tKv,

=

=

(50)

a,u,

a,*a,~,tKtl,

= 1 at I % , ~ K U , ,

. 2

a, = l / ( ~ , t K u , ) ' / ~

do

vi

Definition.

a vector x is

(51)

(52)

uitKui # 0

(53)

v , ~ K v ,=

Sij

. =J

1 i =j, 0 i # j. normalized-more K-normalized. orthonormal I

=0

(29)

V U.

(au

+ bv)

u

V

V,

v

V.

Theorem 1 . If the inner product relation is proper, a subspace U and its orthogonal complement V are disjoint-i.e. have no vector in common, except the null vector.

U

by

XI

, x2 , ..., xk .

U.

U

by (y, y)

V.

(y, xi) y

U

V

6.

105

STRUCTURE OF NORMAL MATRICES

Theorem 2. If U is a proper subspace and V its orthogonal complement under a proper inner product, then V is nonvoid.

U U. U, U,

projection on U ,

U,

V.

V A

U

A (Ax).

U

x

invariant

Theorem 3. If U is invariant for A, then V ,its orthogonal complement under a given inner product, is invariant f o r A#, the adjoint to A under the given inner product.

x

A, Ax

U

U, by

V.

y



=

(xi, Ix,)

xi>

- 1 ) ( X i , Xi) = 0

i

==

(xixi)# 0 by

j,

hi i2

=

1,

10. GENERAL (PROPER) I N N E R P R O D U C T

of on on

<x,Y)

of

(52)

= xtKy

K

(x, x) x y

so

x K-orthogonal, (x, y)

=x

~ yK = o

x <x,AY) =

(53)

y (54)

10.

GENERAL (PROPER) INNER PRODUCT

111

xtKAy

(55)

x

=

(A#x)tKy = x+A#tKy

y KA

K-adjoint

(56)

= A#tK

A A#

= K-'AtK

(57)

K

Definition.

A matrix A is K - n o r m a l AK-lAtK

Definition.

=

if it commutes with its K-adjoint:

A matrix A is K-hermitian A KA

(58)

K-lAtKA

if it equals its K-adjoint:

= K-lAtK

(59)

= AtK

Definition. A matrix A is K-skew-hermitian if A equals the negative of its-K-adjoint:

Definition.

A matrix A is K - u n i t a r y if its K-adjoint equals its

reciprocal: K-lAtK AtKA

(61)

= A-1

-

=K

(62)

Theorem 8. A K-normal matrix A iias a complete set of eigenvectors that can be chosen to be K-orthogonal in pairs, and conversely. Theorem 9. The eigenvalues of a K-hermitian matrix are real, and those of a K-unitary matrix are of unit magnitude.

on so.

112

IV. HERMITIAN, UNITARY, A N D NORMAL MATRICES

As to

K

of

Exercises

1.

a

a

b

2.

2

0

2,

3. (A,

+ jA,)

n x n

A,

A,

A

b,

113

A,

4.

AAt

AtA

A ?

A

A

K

5.

6.

A

7.

A

n x n

II Ax I1 x,

/3

01

(aA + @At)

II x II

=

=

I1 Atx I1

(x, X

Y 2

A ?

8.

A A

(A2 = A), (A2 = A

A

9.

10.

A

At

by

11. A B (AB + BA)

(AB - BA) A

B ?

A

12.

x 13.

A2x = 0, A

Ax

=

0.

B

B A-ABI = O

A

114

I V . HERMITIAN, UNITARY, A N D NORMAL MATRICES

by

p = - xtAx xtBx

p

T2= I

14. A

15. A

?

involution.

At

16.

A

A U

=

U

+ I)(A

-

I)-'

(U - I) A

A (Hint: (A + I) (U I). ?)

+

(A

=

(U - I)-'(U

Cuyley transforms

U

w = (z

(A

-

+ I)

I)-l

+

-

do (U - I)-l

CHAPTER V

Change o f Basis, Diagonalization, and the Jordan Canonical Form Jordan canonical form.

by

by

do

no

by similarity transformation.

change of basis.

1. CHANGE O F BASIS A N D SIMILARITY TRANSFORMATIONS

representation of the abstract vector. As

by

so 115

116

CHANGE OF BASIS, DIAGONALIZATION

by

A on by

? on

x

u.

S. x = su.

u

(1)

x

on on S

on S.

S,

S

u

= s-'x

(2)

y

= AX.

(3)

as

2.

117

EQUIVALENCE TRANSFORMATIONS

A on

by

(1). ? a

x

=

su,

(4)

y = sv

(3),

by S-l,

v = (S-'AS)u

A =

=AU

(5)

A'

S-lAS

(6)

A

(3) S similarity transformation.

by S

similar.

on good

A, A'

on

by no

A

S

A'

by a collineatory

transformation. 2. EQUIVALENCE TRANSFORMATIONS

x (3),

y,

118

CHANGE OF BASIS, DIAGONALIZATION

x

y

A

do do

by x

=

u

su,

=

s-lx

(7)

v = T-’y

y = Tv,

T

S

A equivalence transformation. equivaZent.l

3. CONGRUENT A N D CONJUNCTIVE TRANSFORMATIONS

of

A

B

by B

(’)

=

STAS

A

B

=

STAS

(9)

congruent.

by

B

(10)

conjunctive.

There is some variation in the literature on names for the various types of transformations and on their definitions. We are here following the usage of G. A. Korn M. Korn, “Mathematical Handbook for Scientists and Engineers,” McGraw-Hill, and New York, 1961.

4.

x

119

EXAMPLE

y,

K

by

(x, Y>

x

(1 1)

= xtKy

y,

x on y,

y. (6),

by

(6) (x, y) = (Su)tK(Sv)= uWKSV

(u,v)

x

(12)

(x, y)

y. (u, V)

K

K,

=~

=

K ’ v

(13)

StKS

by by on by

A

by

(3) (1 4. EXAMPLE

As

120

V. CHANGE OF BASIS, DIAGONALIZATION

111, Eq. (36), on

2

111,

-f)

M'

=

S-'MS

= (cos a,

=

a,

cos a, O

;

a,)

(18)

diag(ej7, e-je)

on

5,

11,

by

)

K = - 1( 0 1 2 1 0

K,

Eq. (15) MtKM

K',

=K

(14):

K

= StKS = Z

1 (0 -- 1

(20)

6.

121

INVARIANCE OF THE EIGENVALUES

parity

by on

on u wave basis. on by

5. GAUGE INVARIANCE

by by

by by

no

by

invariant

principle of gauge invariance.

6. INVARIANCE O F T H E EIGENVALUES UNDER A CHANGE O F BASIS

A by A:

x

AX = AX

122

CHANGE

BASIS, DIAGONALIZATION

(4),

A

(4)

by S-l,

u

=

A u =xu A'

Sx

(23) A.

on h

(4)

by

S

by

k

h on

x,

X

x,

2 AX, = AX,

A,

x2

:

+ x1

(24)

(4), XI

=

su, ,

A'u,

x2 =

= h,

+

su,

Ui

A, u,

x1 A'

u,

2

u, .

A' A A'.

7. INVARIANCE O F T H E TRACE

111,

tr A =

12,

Aii

8.

123

VARIATION OF THE EIGENVALUES

S, Bij =

B

z

(27)

(S-'),p4khShj

kh

S-l

z(s-l)iisik = &,

(28)

i

B

of

k 2

3.

8. VARIATION O F T H E EIGENVALUES UNDER A C O NJ UNCTlVE TRANSFORMATION

A of

K (x,x)

x.

K',

K K'

Kt

=

StKS

=

(StKS)t

z=

StKS

K'

=

=K

StKt(St)t

K,

124

CHANGE

BASIS, DIAGONALIZATION

by

K

S

Sylvester's law of inertia.

szgnature

K' on

K

parity matrix

wave basis. 9. DIAGONALIZATION

A. A

..., A,

A,,

x, , ..., x,

:

A

Aj

xi do by

s = (XI

x2

***

x,)

(33)

A,

S S

S by xk .

x1 , ..., x, S

. ?

S

A on

9. 111,

125

15,

wi

xi. witxi

wi

(34)

= Sij

. S. S

S-’

(33))

by

S-1

A by S, AS = A(xl

X,

x,)

by S-l,

= (A1xl

A,x,

**.

A,x,)

(37)

126

V. CHANGE OF BASIS, DIAGONALIZATION

of A, on

reduced to diagonal form. A.

A

A (33)

S-'AS by

S

(33) by S-l

(35),

S A, A,

S x1

=

A,,

(33)

x, w1

x2.

x,

w2, (38),

A

10. DIAGONALIZATION O F NORMAL MATRICES

xi 2

- 8.. 13

S (39)

(39)

3 -

(33)

xi

(34), S-l,

by

(39,

11.

127

THE HERMITIAN MATRIX

by

s:

s-1

=

S A

S

S A

by

=

S

, A,, ...)S-l

=

S

, A,, ...)St

(40)

by

K by

S

S by

11. CONJUNCTIVE TRANSFORMATION O F A HERM I T l A N MATRlX

K by K

K'by S hi

K'. K":

on

K

K'

K" = DtKD sgnAi

A,

=

=

--1 -

K"

K'

=

1

,

A,, ...)

Xi > 0 Ais-l

=

(14)

up

fN(A)

Xi A

***I

=

mi x mi

J,,,i(hi)

Jn

JtJAJ, ...>

=

J,,, ,

JZl(X,)

(15)

by

-+ 00,

k hi. f(x)

off(x)

152

OF A MATRIX

Eq. (1 Eq. Eq. x,, x,

f(x)

f(x) admissable for

by by f(A) =

S

Eq. .-.)>S-l

(16)

by

Eq. f(A)

JnijAi)}

f{Jm,(hl)), f{Jm,(hz)),

=S

**

.)>S-l

mi x mi

\ etc.

/

V.

(4)

4. TRANSMISSION LINE

Eq.

(17)

5.

153

SQUARE ROOT F U N C T I O N

R 0

R2 = (o

e-iRz

Be) = pZI

1

--

= I - jRz

--

(I

+

1

-

p z 2

+

+ j 31!B22R +

2!

(1 - 2!

= I cos Bz

=

2!

1

(19)

1 R2z2 j - R3z3 3!

= I - jRz .

=1

Eq. (3).

1 p4z4 - .:.) +-

4!

1

j-R

Pz

B

-jZ

pz

2

pz

Bz

R.

E(z) = -. E(0)

/3z

E(O) Z(z) = - j Z on

-

jZZ(0)

/3z

flz

+ Z(0) cos fk z

z =0

(21)

of z = 0.

z

e = -p

5. SQUARE ROOT FUNCTION

As

us

z

(22)

154

OF

5. H12= AAt

(23)

H,2

(24)

= AtA

H, ,

H,

of

H,,.

H12

H12

H,,

A x = 0.

of

H12

H,,

x, xtHlzx = x ~ A A ~=x (Atx)t(Atx) = yty

x

y

...,

A?,

A,, A,,

(25)

of H12

...

H12

H12= S{diag(X12,,A,,

...)} S-l

(26)

S{diag(A, , A,, ...)} S-l

(27)

H, H,

=

A, , A,,

S

...

H, go A,, A,,

of

... .

H,

H, H,

U2

A by

A

H,

H, U,

by

(26)

7.

155

EIGENVECTORS

no

A

IV,

H,

H, XII.

6. UNITARY MATRICES AS EXPONENTIALS

As

U

(28)

=

H a

IV)

U

hi

U

=

H

=

ejA2,

...)} S-l

(29)

H ,A 2 , ...)} S-l

(30)

H 10,

S-l

Ht

S

=

,A , , ...)}

=

, A , , ...)} S-l

=

=H

H

hi H

by

A

7. EIGENVECTORS

f(A), A. (4)

156

VI. FUNCTIONS OF A MATRIX

7)

A,

f(hi)

hi

8) f(A).

Theorem 1 . I f f ( x ) is an admissable function of a matrix A, and xiis an eigenvector of A with eigenvalue Xi , then xi is also an eigenvector o f f ( A ) , but with eigenvalue f (A,)..

A A

A f’(A)

A,

1 Theorem 2. If A and B are semisimple, and any eigenvector of A, regardless of how any degeneracies in A are resolved, is also an eigenvector of B, then B can be expressed as an admissable function of A.

A x1

B, A

x2

axl

B

+ /lx2

A,

B A

xi

A

B.

B

hi

/3.

a

Xi.

hi . pi.

pi,

h

pi

hi,

i = i.

A.

=

Xi.

8.

SPECTRUM xk

157

A MATRIX

k # i,

A.

i = k,

B xk

p(A)

A

by Eq.

Lagrange-Sylvester interpola-

tion polynomial

A

B.

A do

go

A, A Eqs.

p’(A,)

p(A,),

=

(dp(A)/dA)+,

p”(A,), ...,pm*-l( Ai).

8. SPECTRUM O F A MATRIX

Definition. If A is a semisimple operator whose distinct eigenvalues are A,, A,, ..., A,, and i f the values of a function f ( h ) are speciJied for h = h,, A = A,, ..., A = h k , then f (A) is said to be defined on the

spectrum of A.

A A A

158

VI. FUNCTIONS OF A MATRIX

Definition. If A is an operator whose distinct eigenvalues are h i , A, , ..., A, , and mi is the length the longest chain with eigenvalue Xi , then (A) is a function such that for all hi ,the set

have specified values, t h e n f ( A ) is said to be defined on the spectrum of A. (A)

g(A) do

A,

on

(35)

f(A)

f(A),

of on

(A)

A.

Eq. A.

on

A Theorem 3. Given a matrix A whose minimum polynomial is of rank m = Erni, there exists a unique polynomial of rank less than m which assumes a specifid nontrivial set of values on the spectrum of A. This polynomial is the Lagrange-Sylvester interpolation polynomial.

#(A), A.

on

~(h), A on m,

?(A)

+(A) #(A) = X

x(A) T(A)

r(A)

(M4

A

A. ~(h),

+ 44

(36)

r(A) m. on

of A, on

,y(A)y(A).

A.

9.

159

EXAMPLE

r(X)

m {r(h) - $(A)} on ~(h)

s(X),

on

m of m

9. EXAMPLE

B As

R,

Eq. (3).

=

e-jRz

(A

B

C D

) -jflx),

x1

x2

+ B = Ze-iBz C Z + D = e-3Bz

AZ

AZ - B

= ZejBz

CZ - D

= -&Bz

B,

D,

A B

=C

C D

=

-jZsinPz -(j/Z)sinPz

=

COSPZ

=

O S ~ Z

160

VI. FUNCTIONS OF A MATRIX

(32) by

of ?

10. C O MM UTATlV lT Y

A =

A

f(A), go

= f ( A ) ,f

(x) A

A A

A

A. Eqs. (17)

I I I.

A Theorem 4. If A and B are semisimple, then they commute if they have a complete set of eigenvectors in common.

if

and only

10.

161

COMMUTATIVITY

2

any

A

B

4

some

xi, y

on

y =

zaixi

Axi

= hixi

B x ~= pixi

AB

y,

=

BA by

A do S-lAS

S-'BS.

on on

A'

=

B

A',

quasidiag(X,I, , h,I, , ...)

=

S-lAS

(41)

hi.

I,

B' B'

=

S-lBS

=

quasidiag(B, , B, , ...)

Bi

Ic.

A"

B" on T-'A'T

by

A"

=

= (ST)-lA(ST)

B"

= T-'B'T = (ST)-'B(ST)

(42)

162

FUNCTIONS

A MATRIX

=

, T, , ...)

T T

Ti

B,.

Ii , ...)

=

,

=

T,, T,,

...

=

A

T, , T, , ...

A'

A"

(ST)

(43)

...)

(44)

B, , B, , ...

B , , ..., A B

a

A on

A

B

11. F U N C T I O N A L RELATIONS

f(x)

A f(x), f(y).

+y)

1 1.

163

eX+Y

= eXeY

(45)

(x)

on by

x

x

providing

do

Theorem 5.

If A and B commute, then, for any positive integer n

(A

+ B)n = $ (3An-kBk k=O

where n! =

k!(n - k)!

n = 1.

(A

n.

+ BY+' = (A + B) 2 (3 k-0

by

A

(47)

164

k

+1

k,

+

2

"

(k - 1)

An-k+lBk

+ Bn+l

2 (" 1') so(" 1'1

= An+l

+

An-k+lBk

+ Bn+l

k= 1

n+l

=

+ 1,

n n

=

2

A

An+l-kBk

by

B do (46):

Theorem 6.

If A and B commute, then eA+B = eAeB

on on

go

k 0

=

n

n = (r r

+ s), n

go

0

03,

k

13. R

165

NOT CONSTANT

A

(48)

do

(4) 12. DERIVATIVE O F e-jR2, R CONSTANT

(4) x(O),

=

-jRe-iRz

(52)

6.

by

+ O(W)

e-jR*2 = I - jR 6z

O(Gz2)

Gz2

1

= lim -{ -jRGz 82-10

6Z

+ 0(6z2)}e-iR2

- -jRe-iRZ

(53) (4)

(52)

13. R NOT CONSTANT

R x(z) = exp (-j

s'

0

R dz) x(0)

(54)

on

166

+ 6z)

z.

by P(z

+ Sz) = d dz

- ep =

+ sz d P ( z ) + O(622) dz 1 {edP/d2Sz - I} 8 6z

81-0

- p deP p dz

dPjdz

J:R dz

R. z.

R

z,

R X.)

R on z R (54).

14. DERIVATIVE O F A P O W E R

of

z,

14.

DERIVATIVE

167

A POWER

by

U”,

n Theorem 7. If U and may be the same, then

V are two n x n matrix functions of x, which

d dU -dz ( U V ) = - Vdz +U-

dV dz

(57)

derivation.

d -(uv> = dz

+ 62)

1

62-0

V(z U(z V(z

dz

=

+ Gz)V(z + 6z) - U(z)V(z) 6z

+ 6x)

+ Sz) = U(z) + 6z U’(z) + ... + 6z) = V(z) + sz V(z) + ...

d

- (UV) =

U(z

6Z+O

1 {(U + 6z U‘)(V+ 6z V’) - UV) 6z

uv + U’V

d -uu” = u u + UU’ dz

only

2UU’, U

U‘.

(59)

U

n dun - - U’Un-1 + UU‘Un-2 +

dz

... + Un-lU‘

(60)

168

VI. FUNCTIONS OF A MATRIX

n

U‘,

nUn-lU’,

n

by

unu-n = 1 dun dz

+ un-dU-” dz

-u-n

= U’

=0

1 1 +(UU’ + U’U) + -(UW’ + UU’U + U’UZ)+ ... 2! 3!

U’

= U’

t

I

+ u + 21!u2+

15. EXAMPLE

by (54) by

Z 0 = fpdz 0

oz) o/z 0

rRdz=( 0 0

-*/

(62)

169

EXERCISES

R. X(2) =

cos

-IZcos

((-jjz)

B

by

(pz). 2,

on

z,

Exercises

1.

d2x dx u-+b-+cx=O dzL dz - d2x + - - + ( 1 -1- dx )=O n2 dz2 z d z 22 d2x dx 22 - - a(a (c) (9- 1) dz2 dz d2x dz2 (b - h2 COS’ Z)X = 0

+

+

+

d

dx

+

*.

&, (P &) + (4

=0

ddX d2x dx ~ + P ~ + q - dz + Y X = O 2. A

3.

=

(:

1 u)

A

=

=0

(54)

170

VI. FUNCTIONS OF A MATRIX

4.

(I

+ A)-'

=I -A

+ A2 - Ad +

?

* * a

of

B=( 5. =

0,

n x n nilpotent.

idempotent.

= =

I,

involution.

6.

X2+ &BX

+ XB)+ C = 0

?

X2+ 5X

+ 41 = 0

7.

3 = y 2 + *Y dz

+q by

1 du Y=-;&

on dY dz

-Y2+PY+Q ?

8. dM _ -- eZAM, dz

M(0) = I

171

EXERCISES

A

X(z)

9.

z

X(u u

+ v) = X(u)X(v) X(z)

w. --

dz

z,

- Ax@)

A (Hint: separately.) (Comment:

u

Polya's equation.

A

10.

X, = ?

n ?

w

CHAPTER VII

The Matricant

-dx(z) -

dz

-jRx(z)

S = -jR,

_-

dx - sx dz

R

S

z.

R on

z,

-

R

2

x(z) = exp(-jRz)x(O)

(2)

by

1. INTEGRAL MATRIX

by 172

R(z)

1.

a

173

INTEGRAL MATRIX

= 0

Theorem. If the given inner product relation is a proper one, then a subspace S, and its orthogonal complement S , is a decomposition of the whole space.

. S, no

S,

u

,u, ,

v: u=u,+v

(v, v)

= (u - U 1 , U - Ul)

ul u on

, by

- u,,)

u, . u,

= uo

+ ciw

>0

,

(23)

u,

.

,

u,

(v, v)

288

XII. SINGULAR A N D RECTANGULAR OPERATORS

w

S,

w (u - u,)

a

u, (u

-

uo - orw,u - uo - aw) = (u - uo ,u

-

uo) - or*(W,u - uo)

- 4 u - uo , w)

3 (u

S, ,

+ I a I2(w,w)

- 110 9 u - UO)

a,

(w,w) # 0. -Kw,u - uo)12 2 0 (w,w> (w,u - uo>= 0

u

- u,

w

S,

.

u

u u

+ (u - uo)

(u - uo) S , .

uoE S ,

Theorem.

= uo

If u

of

w u

of

T, of

ifu

T.

(u,W)

to

u

w,

v.

= (Tv, W) =

(v, T#w)

T#w

w

by

T#

T

u

Tv,

(v, T#w)= 0,

K T#w = 0,

w.

u of

(u,w)

=0

(28)

5.

T#w = 0. u

u

Ty,

-

u,

=

+

= u1

289

EXAMPLE

T

S,

112,

u1 E s, ,

u2 E

S,

s, T,

x

u2

y (U - 111,

Ty)

=

(T#(u - uI), y)

==

0

y T#(u - ~

u

-

u, u

u1

-

S,

u

-

ul

u

(29)

u

(u,u - Ul) = (u - u, , u - Ul)

ul

0

1 = )

+ (Ul , u - Ul)

u1 = u2

S,

(u -u] , u -ul)

=0

u

-

=0

. (30)

u

T

u,.

u

T, on u

by

T#

T#. u.

5. EXAMPLE

T n x n

0 1 0 ...

'

0 0 0 0 0 0

.*.

290

X I I . SINGULAR A N D RECTANGULAR OPERATORS

u=(

0 0 0

T~u =

.**

1 0

('i m

TtT v

=

diag(0, 1, I ,

Ttu,

...,

S, :

by on

6. CONCLUSIONS

T

u = Tv or

u

29 1

EXERCISES

by

Exercises

1.

0

-j

-j 0

-1 j

-1

-1

1

A=(-!

-J

( 1; ) 1

Ax

=

2j - 1

2.

a,

AX = U

(a)

a

= col(1,

-1, -j,j)

a = col(1, 1 , - j , j )

(c) a = col(1 + j , 1 - j , 1 - j , -1

-j)

(d) a.= col(1, 1 - j , -j, 1 - j ) 3. (Caution:

UV~,

?

292

XII. SINGULAR A N D RECTANGULAR OPERATORS

5,

4.

=

uvtK,

K

EX = y x

y

y

5.

Px ?

=a

?

u.

CHAPTER Xlll

The Commutator Operator dW - -- [S,w] = sw - ws dz

S z.

by dx _ - sx dz

x

K,

S W

x

(3)

= xytK

(2),

y dz

do

= @ytK

dz

=

+x

K

SxytK - xytKS

=

=

SxytK

+xyWK

[S, w]

(4)

W’s,

W

293

294

XIII. THE COMMUTATOR OPERATOR

W

.

(l), dM _ -- SM, dz

S

M(0) = I

(5)

z,

W,

W (6), by

W,

W. (5),

S

S

by dSdz

[A,Sl

(7)

A A

Lie book.

1.

295

LIE GROUPS

1. LIE GROUPS

3)

(A, B,...) 1.

I

2.

X

3.

X-l,

not distinct jinite dimensionality.

X

x = X(ff,, a l ,..., a,


f(zn+l)

= f(zn

f(zn-1)

= f(zn - dn) = f ( z n ) - &f’(zn>

dn+l)

f(zn) =

(6)

(7)

f n

n n

d,+,

(5)

=

1

=

fo

N,

fN+l d,

= x1

-

xo

by

f , = f N = 0,

f,’

1

=f2/d,

fl’ = d,u - 2 -fd

f N ’ = -fN-,/dN.

362

XVII. STURM-LIOUVILLE SYSTEMS

where dN+lis taken as appropriate to the boundary condition. We write T

>

fl f 2

fN-1

_- 1 dN+l

f N

1

I

which can be written as

f+' = D+f where

1

=( F ) ( - a i i 7t1

Similarly, from Eq.

+

8i.j-l)

we have 1

fl' = -dl f1

f;

1

=

-(f* -fJ 4 1

fN' =

which we write as -

1 -

d,( f N - f N - l )

0

0

fl f 2

f,

fN

I

\ I

(14)

363

2.

DD+PDf - Qf

-

hRf

D-PD+f - Qf

- hRf = 0

=0

by R-l, Hf

(18)

= hf

H

= R-'(D+PD- - Q)

H

= R-'(D-PD+ - Q)

D-

H. 2. MODIFIED STURM-LIOUVILLE E Q U A T I O N

by g

= p1My1I4f

(21)

Y

w. w

V(w)

w

364

XVII. STURM-LIOUVILLE SYSTEMS

D+D-

d2/dw2

D-D, w, (w7‘ =

1 + h) = gn + hgn’ + 2! h2&’ +

gn+1

=

gn-l

= g(w, - h) = gn - hg,’

(

-

D,

=

h2 I

+ --2!1

h2,”

by 2 1 1 - 2

0 1

-!

A -;

0 0

-2

. Hg

H = D,

= Ag

+ diag(Vl , V, , ...) v,-2 1 0

1 v,-2 1

0 1

v3-2

+

)

(26)

3.

365

THE CHARACTERISTIC EQUATION OF H

no (24).

by 3. T H E CHARACTERISTIC EQUATION O F

H

by

lv1i2-h v z - 2 - x

0 1

I

Vn-2-h

(29)

n x n.

k x k

n -K

by on

V1-2-h

1 pk(h)

1

v2- 2 - h

=(vk-2 vt-,

V1-2-X

-2

1 v*-2-x

!

= ( vk

I

I

-2 -

-pk-2 p,

=

v, - 2 - x

p 2 = (V, - 2

- h)(VZ- 2

-

-

1

V , 's

-h

366

XVII. STURM-LIOUVILLE SYSTEMS

(30)

Jucobi matrix V k

(30) do

V, do by 4. STURM CHAINS V k

p,(h)

p,(X) n,

chain h

pk(h)

=

0,

(a

pk+l(h)

< h < b)

k,

pk-l(h)

k,

po(X)

p,(X),

no

p,;(h)

(30)

by

by

Po(4

k pk(h)

k.

2

pk+l(h)

(30),

h,

p,(h) (30),

A.

p’s Pk

=

=1

pk-l(h)

=

0

h,

A.

pk-2

R(x) (a,b), I s R(h), - co

h

0

n(u) n.

R(x)

+ co,

+ co a

b, pk(a)

k

- 00,

4.

367

STURM CHAINS

pk(h)

pn-Jpn p, pi

b

piPl i # n,

n(b),

i

pnPl(b) > 0,

=

p,(b)

n,

+

0

-

b

by

n(b)

by

p71-l

by

p, , pk(h) pk(A) n(a) = 0. As h +

n(b)

+

up

(30),

by

h -+ - 00,

$),(A)

00,

k

b

odd.

P,~(A)

pn-l/pn P,-~

n. -n.

p,

on

k

n n

k

0 odd,

pk ,

pk-1 .

pk pkPl

A,

$1 n(a)

A

n(b)

V(x) n

(n -

n. V(x)

368

XVII. STURM-LIOUVILLE SYSTEMS

5. R E C U R S I O N FORMULA

(30),

on good

xn = anxn-i

+

n.

b,

a,

(33)

bnxn-z

do do Yn

(34)

= Xn-1

(33) x n = anxn-1

(34)

+

(35)

bnYn-1

(35)

P, = P,

P by

(n -

P,

(36) n. a,

n.

b,

(36) x,

n

.

yn by

odd t i n = Pnpn-1un-2

(38)

5.

369

RECURSION FORMULA

P,P,-, P,

a,,

b,

. (36),

by x,

on x,

x,-,

y , by on n.

1).

XV,

(34),

As Yn = knxn-1

k,

(36)

bn = kanan-,

k

k,

b, ,

a,

.

a,

(33), x, = anxnPl

nna,

=

+ kanan-,xn-,

P,

x,

(44) W,

= wn-1

+ kwn-2 P,

(43).

3 70

XVII.

do

do

Exercises

n x n

n

1.

Jn =

a, b, Jn

c

. (Hint:

3

J, .

(Hint:

a

( I 1)

dt

2.

d,

D, ?

R,

3. R4

+ aR3 + bR2 + CR+ dI = 0 Rn.

(Hint:

R” = fnR3 + gnR2+ h,R

f, , g, , h, , k,

+ k,I

n a =

b

= c =

0, d

=

-1.

CHAPTER X V l l l

Markoff Matrices and Probability Theory

body do up, by

Markofprocess do

by

by

by

N(t). 37 1

372

XVIII. MARKOFF MATRICES AND ‘PROBABILITY THEORY

+ $No

t

=

$N(t)

+ 1).

t,

0, go

Problem of the Gambler’s Ruin.

?

1.

Markoff

random walk

373

STATE VECTOR

drunkard’s walk

1. S T A T E V E C T O R

up

(M

$0

+1

$M

$1

$M, $(M

on,

-

$0. no

on

no

pure state. do

$N

p N,

mixed state.

374

x

probability vector.

on

61.

11 ax 11

=

0111

x 11.

go by

!I x t- y II

= I/ x II

+ II Y II

(3)

2. TRANSITION O R MARKOFF M A T R I X

$n.

$m

pmn

pm,

up

.

m n.

P,~, #m,

by #n

2.

375

TRANSITION OR MARKOFF MATRIX

P,,~,

transition probability

n

m. $n.

do x, $n

pmnx, .

$m ym

$m

(4)

by m by y

= Px

transition

(5)

M a r k 0 8 matrix. s

by x(s).

xk x(s

xk ,

x(s

+

+

P.

There is some confusion in the literature as to just what is called a Markoff matrix. Sometimes pmnis taken as the probability that state m arose from state n, and the Markoff matrix is the matrix of these probability terms. In this case, each row must be a probability vector, since state m must have arisen from some state.

316

XVIII. MARKOFF MATRICES A N D PROBABILITY THEORY

3. EIGENVECTORS O F A MARKOFF MATRIX

xi x(s) x(s)

Eq. ( 5 )

= xi

PXi = xixi x(s

x(s

+ x(s

x(s)

+ 1)

(7)

+ 1) = xixi hi

P no

$M

..., 0)

(8)

xz = col(O,O, ...)0, 1 )

(9)

x1

= col(l,O, 0,

on by

x3,..., x,

up

on ~ ( 0= )

2 i

U a. Xa. - alX1

+ uzx, +

***

(10)

3.

377

m x(m)

=

Cui~imxi

11 x(m) 11 11 xiII

=

1

m.

Xi # 1

=0

(13)

hi 1 hi I > 1,

xi

A’s

m

of x(m)

x(m)

P

I hi I

1, then the matrix can, by a permutation, be put into the f o r m

where the 0’s along the main diagonal are scalars or square matrices.

on no

$0

x

38 1

EXERCISES

1 x 1

(23) on

(n n x 1

x (n -

A A,

by A,

do do

6. CONCLUSIONS

on

do do on

Exercises

$200 $400.

1.

on

p $200 on

$400 ?

$100

(p >

?

$50 ?

?

( p < 8) ?

382

XVIII. MARKOFF MATRICES A N D PROBABILITY THEORY

2.

l b O O O

f !H

A=

'be..

l)

O c a b O O

O

c

a

O

a, b, c

a+b+c=l

P

3.

Q X = a P + ( l -u)Q

0

< < 1.

Q

4.

PQ.

u

5.

uvT, v A

(1 x1 11

Al =

A,, A,, 1, 11 x2 11 = (1 xg I(

=

1

x1 , x2 , ..., x, ,

..., A,, =

.*.= 11 x, 11

=

0,

A

CHAPTER XIX

Stabilitv

by

of

t ?

x(t) = 0,

stable.

x(t) unstable.

x(t)

x(t)

by

admissable. of by

do no

by by

of

x(0). z z

z

383

384

STABILITY

P P

P x ( t ) = exp(Pt)x(O)

(2)

100 do

1. THE BASIC THEOREM FOR STABILITY

Theorem. A system described by Eq. with P constant is stable for all initial conditions if and only if all the eigenvalues of P have negative real parts-i.e., are in the left-hand plane of A.

r(t>= s w S

(3)

(1)

_ du - (S-1PS)u = P’u dt

P P‘

by

(4)

S

P u(t)

M’(t), go

= diag(eAlt,eAzt, ...)u(0) = M(t)u(O)

(5)

1.

THE BASIC THEOREM FOR STABILITY

(4) A 1 0 0

O h 1 0

PI=(.

...

O

385

)

p!. = za pr!, z.+ l = 1

p! . = 0 t.3

(4) U(t) = M(t)u(O)

1

t

0 1 0 0

=o

t2/2! t3/3! t t2/2! 1 t

*")

(7)

j < i

(7')

j

=O

(8)

by

(PM),, .

(5). n t

h

tneat

386

XIX. STABILITY

Definition. P is said to be stable, or a stability matrix if all of its eigenvalues have negative real parts.

2. ROUTH-HURWITZ METHOD

P, P n

h

n

ak = 0 6,

a,

=0

k > 8.

k > *(n

- 1)

b,

p(h)

no Hurwitz matrix

H=

n x n

2.

387

ROUTH-HURWITZ METHOD

by

a,/b,

(bo/co) Routh matrix R :

R

=

b, b2 ...

b, 0

i

c0

C,

0 0 do

...

Theorem. The number of roots of Eq. in the right-hand plane-i.e., with positive real part-is equal to the number of changes of sign in the sequence a, bo do 9

9

9

9

---

Theorem. A polynomial, Eq. (9) has all of its eigenvalues in the lefthand plane-i.e., with negative real part-if all of the terms a , , b, , c, , ... are nonzero and have the same sign.

b, , c, , d o , by regular

by

E

no (A2

+h +

+ h - 1) = A4 + u3 + 2 0

0

0 1

1 - 1

H=(i

A2 -

1

=0

388

XIX. STABILITY

1, 2,

1, 2,

-1

of of A4

+

a 3

- h2 -

u- 3 =

(A2

+ h - 3)(h2 + h + 1) = 0

2-2

co 2-2

0

0

0

2-2

0

0

0 a-3

2 -2

(+

=

0

0 -2

O

O

+ - - -).

+ 6/a -3

B

B,

0.

co = E 2 -2

0

1, 2, c

0

0

0 0

+ b / B , -3 (+ + + + -).

-2

0

0 -2 0 0

+ 6/a -3

4.

CRITERION OF LIENARD A N D CHIPART

389

on do by

3. H U R W I T Z D E T E R M I N A N T S

dk,

k

n): A,

= b,

A,

=

Ai>O,

bo bl b, a, U , a2 10 b, b,l

i 0, > 0,

> 0, > 0,

a,-,,-, an--2k-l

> 0, > 0,

A,,-, > 0, A,, > 0, A 2 k - l > 0, A,, > 0,

no

on 5. LYAPUNOV’S SECOND METHOD

by

< k < n/2 1 < k < n/2 1 < k < n/2 1 X,>O xtx

L

A,

xtMx xtx