Kronecker Products and Matrix Calculus: With Applications (Mathematics and Its Applications)

Kronecker Products and Matrix Calculus: with Applications ALEXANDER GRAHMvI, M.A., M.Sc., Ph.D., C.Eng. M.LE.E. Senior Lecturer in Mathematics, The Open University, Milton Keynes

.,...;.

ELLIS HORWOOD LIMITED Publishers· Chichester

Halsted Press: a division of JOHN WILEY & SONS New York· Brisbane· Chichester· Toronto

first published in 1981 by ELLiS HORWOOD LIMiTED Market Cross House, Cooper Street, Chichester, West Sussex, PO 19 lEB, England 11Ie publisher's colophon is reproduced from James Gillison's drawing of the allcient Market Cross, Chichester.

Distributors: Australia, New Zealand, South-east Asia; Jacaranda-WUey Ltd., Jacaranda Press, JOHN WILEY & SONS INC., G.P.O. Box 859, Brisbane, Queensland 40001, Australia Canada: JOHN WILEY & SONS CANADA LIMITED 22 Worcester Road, Rexdale, OntariO, Canada. b'urope, Africa.' JOHN WILEY & SONS LIMITED Baffins Lane, Chichester, West Sussex, England, North and South America and the rest of the world: Halsted Press: a division of JOliN WILEY & SONS 605 Third Avenue, New York, N.Y. 10016, U.S.A.

© 1981 A. Graham/Ellis Horwood Ltd. British Library Cataloguing in Publication Data Grw:un. Alexander Kronecker products and matrix calculus. (Ellis Horwood series in mathematics and its applications) 1. Matrices 1. Title 512.9'43 QA188 Library of Congress Card No. 81-7132

AACR2

ISBN 0-85312-391-8 (Ellis Horwood Limited, Library Edition) [SBN 0-85312-427-2 (Ellis Horwood Limited. Student Edition) ISBN 0-470-27300-3 (Halsted Press) Typeset in Press Roman by Ellis Horwood Ltd. PIlnted in Great Britain by R. J. Acford, Chichester

COI'YRIGIIT NOTICE -

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Table of Contents Author's Preface ..........................................7 Symbols and Notation Used ..................................9 Chapter 1 - Preliminaries 1.1

Introduction ....................................... 11

1.2 Unit Vectors and Elementary Matrices ...................... 11

1.3 Decompositions of a Matrix ............................. 13

1.4 The Trace Function .................................. 16 1.5 The Vec Operator

................................. 18

.

Problems for Chapter I ................................20

.01.

Chapter 2 - The Kronecker Product

2.1 Introduction ....................................... 21 2.2 Definition of the Kronecker Product .......................21 (/]

2.3 Some Properties and Rules for Kronecker Products ............. 23

2.4 Definition of the Kronecker Sum .........................30 2.5 The Permutation Matrix associating vccX and vecX' ............. 32 4-.

Problems for Chapter 2 ................................ 35

Q[>

a..

Chapter 3 - Some Applications for the Kronecker Product

3.1 Introduction ....................................... 37 3.2 The Derivative of a Matrix ..............................37 0..

v-1

3.3 Problem 1: solution of AX + XB = C ..................... 38 3.4 Problem 2: solution of AX + XA = µX ..................... 40 3.5 Problem 3: solution of X = AX + XB ..................... 41 3.6 Problem 4: to find the transition matrix associated with }..

the equation X = AX + XB ............................ 42

3.7 Problem 5: solution of AXB = C .........................44 3.8 Problem 6: Pole assignment for a Multivariable System...........45

380A

Table of Contents

6

Chapter 4 - Introduction to Matrix Calculus

4.1 Introduction ....................................... 51

4.2 The Derivatives of Vectors ............................. 52 4.3 The Chain rule for Vectors ............................. 54 4.4 The Derivative of Scalar Functions of a Matrix

with respect to a Matrix ............................... 56 its Elements and Conversely ............................60 4.6 The Derivatives of the Powers of a Matrix ................... 67 Problems for Chapter 4 ................................ 68 4.5 The Derivative of a Matrix with respect to one of

Chapter 5 - Further Development of Matrix Calculus including an Application of Kronecker Products

5.1 Introduction ....................................... 70 5.2 Derivatives of Matrices and Kronecker Products ............... 70

5.3 The Determination of (avecX)/(avecY) for more

complicated Equations ............................... 72 5.5 The Matrix Differential ................................ 78 Problems for Chapter 5 ................................ 80 5.4 More on Derivatives of Scalar Functions with respect to a Matrix .... 75

A.°

Chapter 6 - The Derivative of a Matrix with respect to a Matrix

6.1 Introduction ....................................... 81 6.2 The Definition and some Results ......................... 81 ...

6.3 Product Rules for Matrices ............................. 84 6.4 The Chain Rule for the Derivative of a Matrix with respect to Matrix .88

Problems for Chapter 6 ................................ 92

Chapter 7 - Some Applications of Matrix Calculus '_'

7.1 Introduction ....................................... 94 7.2 The Problems of Least Squares and Constrained Optimization in Scalar Variables ..................................... 94 'C7

7.3 Problem 1: Matrix Calculus Approach to the Problems

of Least Squares and Constrained Optimization ................96 7.4 Problem 2: The General Least Squares Problem ............... 100 ...

'v,

7.5 Problem 3: Maximum Likelihood Estimate of the Multivariate Normal 102

7.6 Problem 4: Evaluation of the Jacobians of some Transformations... 104 7.7 Problem 5: To Find the Derivative of an Exponential

Matrix with respect to a Matrix ......................... 108

Solution to Problems ..................................... III Tables of Formulae and Derivatives ............................ 121

Bibliography ........................................... 126

Index ............................................... 129

Author's Preface

'i.

...

_r.

My purpose in writing this book is to bring to the attention of the reader, some recent developments in the field of Matrix Calculus. Although some concepts, such as Kronecker matrix products, the vector derivative etc. are mentioned in a few specialised books, no book, to my knowledge, is totally devoted to this subject. The interested researcher must consult numerous published papers to appreciate the scope of the concepts involved. Matrix calculus applicable to square matrices was developed by Turnbuil [29,301 as far back as 1927. The theory presented in this book is based on the works of Dwyer and McPhail [15] published in 1948 and others mentioned in the Bibliography. It is more general than Turnbull's development and is applicable to non-square matrices. But even this more general theory has grave limitations, in particular it requires that in general the matrix elements are non constant and +~+

O..

rte.

''.

i1.

T°°

4-.

.'7

independent. A symmetric matrix, for example, is treated as a special case.

~a.,

1)¢

CV]

.'7

'r+

Methods of overcoming some of these limitations have been suggested, but I am not aware of any published theory which is both quite general and simple enough to be useful. The book is organised in the following way: Chapter 1 concentrates on the preliminaries of matrix theory and notation which is found useful throughout the book. In particular, the simple and useful

elementary matrix is defined. The vec operator is defined and many useful

l/1

..^

'-'

^.N

...

in.

...

9a)

relations are developed. Chapter 2 introduces and establishes various important properties of the matrix Kronecker product. Several applications of the Kronecker product are considered in Chapter 3. Chapter 4 introduces Matrix Calculus. Various derivatives of vectors are defined and the chain rule for vector differentiation is established. Rules for obtaining the derivative of a matrix with respect to one of its elements and conversely are discussed. Further developments in Matrix Calculus including derivatives of scalar functions of a matrix with respect to the matrix and matrix differentials are found in Chapter 5. '27

Chapter 6 deals with the derivative of a matrix with respect to a matrix.

Author's Preface

8

'..

This includes the derivation of expressions for the derivatives of both the matrix product and the Kronecker product of matrices with respect to a matrix. There is also the derivation of a chain rule of matrix differentiation, Various applications of at least some of the matrix calculus are discussod in Chapter 7, By making use, whenever possible, of simple notation, including many worked examples to illustrate most of the important results and other examples at the end of each Chapter (except for Chapters 3 and 7) with solutions at the 4..

a`1

1..

4.n 'CJ

461

end of the book, I have attempted to bring a topic studied mainly at post-S7

'C1

graduate and research level to an undergraduate level.

Symbols and Notation Used A,B,C... matrices

el e

Ell 0,,, SU

A., Aj. A1.'

(A')., (A').; tr A

vecA AOB iff diag {A} 8Y

the transpose of A the (i, j)th element of the matrix A the matrix A having arf as its (4 j)th element the unit matrix of order m X in the unit vector the one vector (having all elements equal to one) the elementary matrix the zero matrix of order in X m the Kronecker delta the lth column of the matrix A the jti row of A as a column vector the transpose of Af. (a row vector) the ithe column of the matrix A' the transpose of the ith column of A' (that is, a row vector) the trace of A an ordered stock of columns ofA the Kronecker product of A and B if and only if the square matrix having elements all, a22, . . . along its diagonal and zeros elsewhere .w,

ari [aif] I,,,

,..

A'

a matrix of the same order as Y

aXrs

ayf

ax Ers

E#

a matrix of the same order as X

an elementary matrix of the same order as X an elementary matrix of the same order as Y

CHAPTER I

Preliminaries

1.1 INTRODUCTION

In this chapter we Introduce some notation and discuss some results which will

be found very useful for the development of the theory of both Kronecker products and matrix differentiation. Our aim will be to make the notation as

fl.

.".

simple as possible although inevitably it will be complicated. Some simplification may be obtained at the expense of generality. For example, we may show that a result holds for a square matrix of order n X n and state that it holds in the more general case when A is of order in X n. We will leave it to the interested reader to modify the proof for the more general case. Further, we will often write L°.

.....

or

or

justDij instead of

m n

ij i=1 j=1

when the summation limits are obvious from the context. Many other simplifications will be used as the opportunities arise. Unless of particular importance, we shall not state the order of the matrices considered. It will be assumed that, for example, when taking the product All or ABC the matrices are conformable. 1.2 UNIT VECTORS AND ELEMENTARY MATRICES

The unit vectors of order n are defined as 1

0

0

0

1

0

,

e2 = 0

,

..., e _

0

...

e1 = 0 Pi

L0J

L1

Preliminaries

12

[Ch. 1

The one vector of order n is defined as 11 1

e=

(1.2)

1

1

From (1.1) and (1.2), obtain the relation

e = Eel

(1.3) r.,

The elementary matrix E,i is defined as the matrix (of order m X n) which has a unity in the (i, f)th position and all other elements are zero. For example,

000...0

001 ...0 E23 =

000...0

(1.4)

Lo00...0J The relation between e1, ei and E11 is as follows

Eli = ei el

ti.

Cs.

where ei denotes the transposed vector (that is, the row vector) of el.

t27

Example 1.1 Using the unit vectors of order 3

(i) form Ell, E21, and E23 (ii) write the unit matrix of order 3 X 3 as a sum of the elementary matrices. Solution 1

1

E11= 0 [1 001= 0

0

1 [100]= 100 0 0

E23 =

000 000 --+

E21=

1

00

000

1

0

000 0 0 0

[0 0 1]= 0 0

..-

(i)

1

000

Decompositions of a Matrix

Sec. 1.3]

13

3

(ii)'

! = Eit + E22 + E33 =

eiej r= The Kronecker delta Sij is defined as

1ift=/ Oifizkj

Sid CAD

it can be expressed as

Sij=ejei=ejei

(1.6)

.

We can now determine some relations between unit vectors and elementary matrices.

Eijer = eiejer (by 1.5) (1.7)

= 5/rei and

e,.Eii = e.eiej (1.8)

= Sriej Also

EijErs = eieieres = 5jetes = SjrEis

(1.9)

In p articular if r =f, we have

EijEjs=51jEis=Eis and more generally (1.10)

LijEjsEsrn = EisEsm = Eim

Notice from (1.9) that

EijErs = 0 if / # r . 1.3 DECOMPOSITIONS OF A MATRIX We consider a matrix A of order m X n having the following form

A

all all

ainn

a12

a22

.

a2n

= [a11]

Lamlamt amnJ We denote then columns of A by A.1, A.2, ... A,n. So that

A.j = a2i an,j

(j = 1, 21 .... n)

(1.12)

[Cll. 1

Preliminaries

14

and them rows of A by A1., A.2, ...A.. so that au

A. =

(i = 1,2,... ,m)

ate

(1.13)

at,

Both the A.l and the A. are column vectors. In this notation we can write A as the (partitioned) matrix or as

A = [A.1 A.2 ... A.,,]

(1.14)

A = [A1.A2.... A,,,.]'

(1.15)

fl,

(where the prime means 'the transpose of'). For example, let

A

all al a21 a22

so that

A1. =

all

and

a21

A2. _

a12

then

a22

Pall

a2I' = call a121 a221

L 12

=A.

La21 a22

The elements, the columns and the rows of A can be expressed in terms of the unit vectors as follows:

So that

The jth column A.1 = Ael

(1.16)

The ith rowAi '= ejA.

(1.17)

A;. = (e,A)' = A'e1.

(1.18)

The (i,j)th element ofA can now be written as

all = ejAel = eeA'el We can express A as the sum

(1.20) '+f

A = EEailEfl A = EEaile,e1.

0

(where the Ell are of course of the same order as A) so that

(1.21)

--*

1(,

Decompositions of a Matrix

Sec. 1.31

15

From (1.16) and (1.21)

A. j = Aej = (2Eaiieie)ei = ZEatjet(e/ej) (1.22)

= 2;a;ie; . Similarly

so that

At. = Ea;jej

(1.23)

j

A;. = Eatjej

(1.24)

.

I

It follows from (1.21), (1.22), and (J.24) that

A = XA.jej

(1.25)

A = Eet A;.' .

(1.26)

GIN

[1.

and

Example 1.2 Write the matrix

Fall a,2 N..

A=

L2l a2J as a sum of: (i) column vectors of A; (ii) row vectors of A.

Solutions (i) Using (1.25)

A = A.le'1 + A.2e2

[

a

21

[1 03 +

[0 1]

a22

al e

Using (1.26)

A = el A1: + e2A2.'

ro [all a12] + [00

[a,21

a,22]

a.)

There exist interesting relations involving the elementary matrices operating on the matrix A. For example

EtjA = e;ej'A = e1Aj '

(by 1.5) (by 1.17)

(1.27)

[Ch. I

Preliminaries

16

similarly

AErj = Ae;ej' = A.ree

.(by 1.16)

(1.28) (1.29) CST

.On

sa that

AEij = A.jee AE,jB = Aejej'B = A.,B1.'

(by 1.28 and 1.27)

,ErjAEr,i = ere/Aeres

(by 1.5)

= ejalre'l

(1.30)

(by 1.19) (1.31)

= ajreie; = airEls +:.

t.)

In particular (1.32)

Ej1AErr = airEir

v.. ,...

Example 1.3 Use elementary matrices and/or unit vectors to find an expression for (i) The product AB of the matrices A = [a,1] and B = [bij]. (ii) The kth column of the product AB (iii) The kth column of the product XYZ of the matricesX= [xji], Y= and Z = [zii] Solutions

(i) By (1.25) and (1.29)

A = EA. i e, = EAEii hence

AB = E(AE11)B = E(Aej)(ej'B) = EA.1Bj.'

(by (1.16) and (1.17)

(ii) (a) by (1,16)

`s]

(AB).k = (AB)ek = A(Bek) = AB.k (b) From (i) above we can write

(AB).k =

E(Aejej'B)ek = E(Aej)(e%Bek)

= EA./bjk i

(iii)

by (1.16) and (1.19)

(XYZ).k = Ezjk(XY).j

by (ii)(b) above

= E(zjkX)Y.j

by (ii)(a) above.

1.4 THE TRACE FUNCTION The trace (or the spur) of a square matrix A of order (n X n) is the sum of the diagonal terms

n

art 1=1

The Trace Function

Sec. 1.4] We write

17

tr A = Eau

(1.33)

From (1.19) we have

aj1 = e';Aet, so that

tr A = Ee'iAei

(1.34)

From (1.16) and (1. 34) we find

tr A = Ee'iA.j

(1.35)

and from (1.17) and (1.34)

tr A = EAj.'ej

(1.36)

.

We can obtain similar expression for the trace of a productAB of matrices. For example

tr AB = Ee'jABej

(1.37)

t

= EE(e'Ae1)(e%Bet) II

= Efatlbfj

(See Ex. 1.3) (1.38)

Similarly tr BA = EeeBAe1

=

= EEbljat/

(1.39)

From (1.38) and (1.39) we find that

trAB=trBA.

(1.40)

From (1.16), (1.17) and (1.37) we have

tr AB = EA; B.t

(1.41)

Also from (1.40) and (1.41)

tr AB = EB1.A.j Similarly

.

(1.42)

tr AB' = EAj.B1 .

(1.43)

and since tr AB' = Is A'B

tr AB' = EA.'jB.t

(1.44)

Preliminaries

18

[Ch. I

'C3

Two important properties of the trace are .nd

tr (A + B) = tr A + tr B

(1.45)

tr (a A) = a trA

(1.46)

where a is ascalar.

These properties show that trace is a linear function. For real matrices A and B the various properties of tr (AB') indicated above show that it is an inner product and is sometimes written as

tr (AB') _ (A, B) 1.5 THE VEC OPERATOR We shall make use of a vector valued function denoted by vec A of a matrix A defined by Neudecker (221. If A is of order m X n A.1

vecA = A.2 LA. J,

(1.47) .

-U.

From the definition it is clear that vecA is a vector of order mn. For example if

A =

a21 azz C11 a'2

then

rai n

vecA = a21 a12

a22

Example 1.4 Show that we can write tr AB as (vec A')' vec B Solution By (1.37) tr AB =

Ee'jABe1

= EAi;B,1

by (1.16) and (1.17)

(since the ith row of A is the ith column of A')

Sec. 1.51

The Vec Operator

19

Hence (assuming A and B of order n X n) B.l

tr AB = E(A').1'(A').i 2'.

_ (vec A')'vec B

(A').,,']

B.2

B,

Before discussing a useful application of the above we must first agree on notation for the transpose of an elementary matrix, we do this with the aid of an example. X11 Xl2 X13

Let X =

X21 X22 X23

then an elementary matrix associated with will X will also be of order (2 X 3). For example, one such matrix is

_ E12=

0

0

1

000

The transpose of E12 is the matrix 0 0

E12 =

0

1

00 'C7

t3.

..,

.N.

c$'

Although at first sight this notation for the transpose is sensible and is used frequently in this book, there are associated snags. The difficulty arises when the suffix notation is not only indicative, of the matrix involved but also determines specific elements as in equations (1.31) and (1.32). On such occasions it .NJ

..d

will be necessary to use a more accurate notation indicating the matrix order and the element involved. Then instead of E12 we will write E12(2 X 3) and instead of E12 we write E21(3 X 2),

More generally if X is a matrix or order (in X n) then the transpose of Ers (171 X n)

will be written as Ers

unless an accurate description Is necessary, in which case the transpose will be written as

Esr(nXm)

.

Now for the application of the result of Example 1.4 which will be used later on in the book.

[Ch. 1]

Preliminaries

20

From the above tr E,''A = (vec Ers)' (vec A) ars

..y

where ars is the (r,s)th element of the matrix A. We can of course prove this important result by a more direct method.

tr E',.sA = Ee ErsAek ai/ekese;.eiejek

(sinceA =>aiiEij)

i, j, k

ij'k$Sri'jk = ars i,1, k

Problems for Chapter 1

(1) The matrix A Is of order (4 X n) and the matrix B is of order (n X 3). Write

the product AB in terms of the rows of A, that is, A,., A2., .. , and the columns of B, that is, B.1, B.2, ...

.

(2) Describe in words the matrices (a) AEik

and

(b) EikA .

(3)

Show that

(a) trABC= EA1.BC.i

(b) trABC= trBCA=trCAB

.-.

C.1

Write these matrices in terms of an appropriate product of a row or a column of A and a unit vector.

Show that tr AEij = aji

B = [bij] is a matrix of order (n X n) diag {B} = diag {bll, b22, ... , b,,,, } = EbiiEii .

Show that if aij = tr BEjj6jj

then A = [aij] = diag{B}

CHAPTER 2

The Kronecker Product 2.1 INTRODUCTION

Kronecker product, also known as a direct product or a tensor product is a concept having its origin in group theory and has important applications in +.,

particle physics. But the technique has been successfully applied in various fields

of matrix theory, for example in the solution of matrix equations which arise when using Lyapunov's approach to the stability theory. The development of the [3.

technique in this chapter will be as a topic within the scope of matrix algebra.

2.2 DEFINITION OF THE KRONECKER PRODUCT

Consider a matrix A = [aqj of order (m X n) and a matrix B = [bq] of order

.''

(r X s). The Kronecker product of the two matrices, denoted by A O B is defined as the partitioned matrix

AOB =

a11B

a12B

...

a21B

a22B

...

a2,B a,r,,,

LamIB

(2.1)

B

A O B is seen to be a matrix of order (rnr X its). It has inn blocks, the (i,j)th block is the matrix a11B of order (r X s). For example, let

A E P' 11 ail f a21

B_ I bil b121

a221

I b21 b22

then

rallbll allbl2 al2bll a12b12

AOB

a11B a12B

La21B a22B

=

a11b21 a1lb22 a12b21 a12b22 a21b11 a21b12 a22b11 a22b12 a21 b21 a21 b22 a22 b21 a22 b22

(Ch. 2

The Kronecker Product

".. Z2

CUD

Notice that the Kronecker product is defined irrespective of the order of the makes involved. From this point of view it is a more general concept than matrix multiplication. As we develop the theory we will note other results C3.

w{..'

which are more general than the corresponding ones for matrix multiplication.

The Kronecker product arises naturally in the following way. Consider two

linear transformations

x = Az and y = Bw

xt

x2

Fall

at2 r a t

Last

a22

Yt

btr

Y2

bet

and

Z2

._._

which, in the simplest case take the form bb2r22,

wt (2.2)

LW J

We can consider the two transformations simultaneously by defining the following vectors

xiyt

ztwt

XI VI

z ws

x 0y =

v= z© w=

and

(2.3)

I x2Yt

z2wt

x2 Y2

z2w2

.

To find the transformation between µ and v, we determine the relations between the components of the two vectors. For example, xtyt = (attzt + at2z2) (btt wt + bt2w2) = all btt (ziwt) + all bt2(ztw2) + at2btt(z2wt) + at2bt2(z2w2)

attbt2

at2btt

a,-2b,2

attb21

all b22

at2b2t

a12b22

a2tbtt

a21b12

a22brt

a22b12

a21b12

a2tb22

a22b2t

a22 b22

.N..

u=

alibi,

tr.

Similar expressions for the other components lead to the transformation

v .N.

or

µ = (A®B)v, that is

Az®Bw = (A®B)(z(Dw)

.

(2.4)

Example 2.1

Let Eq be an elementary matrix of order (2 X 2) defined in section 1.2 (see 1.4). Find the matrix 2

U=

2

Ej, i ®EI I

Sec. 2.3]

Some Properties and Rules for Kronecker Products

23

Solution

L°-!+

U =Ell (8) Ell +E1,2 ®E2,1 +E11 ® E12 +E2,2 ®E2,2

f11 ® r61 + roa1 ® roof + (001 00

0 0

0 0

1

0

I

of

(lo

'

+ ( 011 ®

so that 1

U =

it

Lo 0J

011

0 0 0

0 0

1

0

0 0

1

0 0 0

0

1

Note. U is seen to be a square matrix having columns which are unit vectors er(i = 1, 2,.. ). It can be obtained from a unit matrix by a permutation of rows or columns. It is known as a permutation matrix (see also section 2.5). 2.3 SOME PROPERTIES AND RULES FOR KRONECKER PRODUCTS

We expect the Kronecker product to have the usual properties of a product. I

If a is a scalar, then

A O (aB) = a(A ®B)

.

(2.5)

Proof The (i, j)tli block of A O (aB) is [are (aB)J

= a[a11BJ

= a[(i, j) th block of A O BJ The result follows.

It The product is distributive with respect to addition, that is (a)

(b)

(A+B)OC = AOC+B®C A®(B+C) = ul ®B+A®C

Proof We will only consider (a), The (i, j)th block of (A + B) ® C is (ali + b1i) C

.

The (i, j)th block of A ® C + B ® C is

a11C+b;1C = (a11+bl)C

(2.6)

(2.7)

(Ch. 2

The Kronecker Product

24 Since the two

-III The

blocks are equal for every (i,j), the result follows.

product is associative

A®(B®C) _ (A(2-9 B)®C .

(2.8)

IV There exists a zero element Ornr, = Orr, 2) On (2.9)

a unit element Imn ° Im ® In 0

The unit matrices are all square, for example In, in the unit matrix of order (jn X m).

Other important properties of the Kronecker product follow. (2.10)

V (A ®B)' = A' ®B' Proof '+7

The (i,j)th block of (A (D B)' is ai jB' .

VI (The `Mixed Product Rule').

(A ®B) (C ®D) = AC ®BD

(2.11)

provided the dimensions of the matrices are such that the various expressions exist.

C.,

Proof The (i,j)th block of the left hand side is obtained by taking the product of the t.,..

ith row block of (A ® B) and the /th colum block of (C ® D), this is of the following form c11D

(ai1B ai2B ... ajnB)

c21D

cn1D

= EajrcriBD . r

-

-

The (i, j)th block of the right hand side is (by definition of the Kronecker product) gj1BD

where gji is the (i, j)th element of the matrix AC. But by the rule of matrix multiplications

gji=Zajrcri

Sec. 2.31

Some Properties and Rules for Kronecker Products

25

Since the (i,j)th blocks are equal, the result follows.

VII Given A(m X m) and B(n X n) and subject to the existence of the various inverses,

(A©B)'' = A"' OBy'

(2.12)

Proof Use (2.11 )

(A ®B) (A-' ®B"') = AA-' ®BY-' = I, ®In = Inv. The result follows.

VIII (See (1.47))

vec(AYB) _ (B' ®A) vec Y

(2.13)

Proof We prove (2.13) for A, Y and B each of order n X n. The result is true for A(m X n), Y(n X r), B(r X s). We use the solutions to Example 1.3(iii).

(AYB).k = E(bikA)Y.i

'-

i

-,

Y.1

_ [blkA b2kA ... bnkA1

Y. 2

Y.n

= [B.k'®A]vecY = [(B')k: ®A] vec Y since the transpose of the kth column of B is the kth row of B'; the results follows.

Example 2.2 Write the equation

all a21

a12

XI

X3

a22

X2

X4

X11

C12

X21 `2J

in a matrix-vector form.

Solution The equation can be written as AXI = C. Use (2.12), to find

vec (AXI) = (1®A) vec X = vcc C ,

The Kronecker Product

26

[Ch. 2

so that

0-1

x1

C1t

0

x2

X21

a12

X3

a12

o`"

x4

Lc22

Fall a12 0 a21

a22 0

0

0

0

0

all

a22

a21

Example 2.3 A and B are both of order (n X n), show that

(i) vecAB=(1®A)vecB (ii) vecAB=(B'®A)vecl (iii) vec AB = E (B').k ® A.k Solution

(1) (As in Example 2.2)

In (2.13) let Y = B andB =1. (ii) In (2.13) let Y = I . (iii) In vec AB = (B' ®A) vec I substitute (1.25), to obtain

f`7

vecAB =

[(B').ie; O EA.lei]vecl

= [((B').i®A.J)(e.® ee)]

ij

vec 1

(by 2.11)

The product e', O ei' is a one row matrix having a unit element in the [(i - 1)n + j]th column and zeros elsewhere. Hence the product ,U.

[(B').; ®A.i] [el' O el] is a matrix having

(B').1®A.1

'fl

as its [(i -1)n + j]th column and zeros elsewhere. Since vecl is a one column matrix having a unity in the 1st, (n + 2)nd, (2n + 3)rd . . . n2rd position and zeros elsewhere, the product of 61.

[(B').I ®A.l] [ej ® e)] and vec I is a one column matrix whose elements are all zeros unless i and j satisfy

(i-1)n+j = l,orn+2,or2n+3,...,orn2

Sec. 2.3j

Some Properties and Rules for Kronecker Products

that is

1=j=1 or

27

or i=j=3 or ..., i=j=n

i=j=2

in which case the one column matrix is

(B').i®A.r (i = 1,2,...,n) The result now follows.

IX If (X;} and (xj) are the eigenvalues and the corresponding eigenvectors for A and (µi} and (yi) are the eigenvalues and the corresponding eigenvectors for B, then

A®B has eigenvalues (Xrµj} with corresponding eigenvectors (xi ® yi}.

Proof By (2.11)

(A ® B) (x, ® yi) _ (Ax,) © (Byi) _ (Xixr) ® (µ1y1)

= Xjµi(x1 ®yj)

(by 2.5)

The result follows.

X Given the two matrices A and B of order n X n and m X m respectively JAOBI = IAImJBV" where IAA means the determinant of A.

Proof Assume that X1, X2, ... , X and µr, µ2, ... , µ,,, are the eigenvalues of A and B respectively. The proof relies on the fact (see [18] p. 145) that the determinant of a matrix is equal to the product of its eigenvalues. Hence (from Property IX above)

IAOBI = jjXjuf i,l n

n

X ' II µj) 1x2 tI P) 1=t

(X1 X2

...

JAI"' IBI°

...

1=t

ll(22 ...

rr

t X nr1=tl l µ//

The Kronecker Product

28

[Ch. 2

Another important property of Kronecker products follows.

AOB = Ut(BOA)U2

(2.14)

where U1 and U2 are permutation matrices (see Example 2.1).

Proof Let AYB' = X, then by (2.13) (BOA) vec Y = vecX X.

(1)

on taking transpose, we obtain

BY;t' = X' So that by (2.13)

(A 0 B) vec Y' _ vecX'

(2)

.

'I]

From example 1.5, we know that there exist permutation matrices U1 and U2 such that

vec X' = U1 vec X

and

vec Y = U2 vec Y'

.

Substituting for vec Yin (1) and multiplying both sides by U1, we obtain

U1(B 0A)U2vecY' = U1 vecX

.

(3)

Substituting for vec X' in (2), we obtain

(A O B) vec Y' = U1 vecX

.

(4)

The result follows from (3) and (4). We will obtain an explicit formula for the permutation matrix Uin section 2.5. Notice that U1 and U2 are independent of A and B except for the orders of the matrices.

XII if f is an analytic function, A is a matrix of order (n X n) and f(A) exists, then

f(1,n&A) = Im ID AA) and

f(A O Im) = f(A) O I. Proof Since f is an analytic function it can be expressed as a power series such as

f(z) = a°+a1z+a2z2+.. so that

f(A) = aoI,, +a1A+a2A2+... _

where A° = I.

u^,.

By Cayley Hamilton's theorem (see [18]) the right hand side of the equation for f(A) is the sum of at most (n + 1) matrices.

Sec. 2.3]

Some Properties and Rules for Kronecker Products

29

We now have

k =O

k=0

k=0 err,

®

7a

k

k=0

Im ©f (A)

'"1

This proves (2.15); (2.16) is proved similarly. We can write

f(A (D I,,)

)'ak(A Ox Im)k k -O

(Ak ©Im)

by (2.11)

k=0

akAk ®lm) k=0

akA®0Im

=

by (2.6)

k=0

f(A) (& Irn

This proves (2.16). An important application of the above property is for

f(z) = eZ

.

(2.15) leads to the result

elm 6A = Im O eA and (2.16) leads to eA ®rm = eA

O It n

Example 2.4 Use a direct method to verify (2.17) and (2.18).

elm®A =

a~'

Solution

(2.17)

(2.18)

The Kronecker Product

30

[Ch. 2

The right hand side is a block diagonal matrix, each of the m blocks is the sum

I,,,+A+21 A2+...

= eA

.

The result (2.17) follows. eA®Im

(In®Im)+(A(D Im)+21 Q. ®A)2+... ( 1 n ®In,) + ( A ®1m) + 1(A2 01m) + .. .

= Q,,+A+2A2+...)OOIm = eA ®I,,,

XIII tr(A®B)=trAtrB Proof Assume that A is of order (n X n)

tr(A®B) = tr(a1,B)+tr(a22B)+...+tr(annB) = a11trB+a22trB+...+anntrB = (all +a22+...+a.... )trB = tr A tr B . 2.4 DEFINITION OF THE KRONECKER SUM Given a matrix A(n X n) and a matrix B (m X m), their Kronecker Sum denoted by A ®B is defined as the expression

AG+B = A©I,,+1n®B

(2.19)

We have seen (Property IX) that if {X;} and {pj} are the eigenvalues of A and B respectively, then {X;pj} are the eigenvalues of the product A ® B. We now show the equivalent and fundamental property for A (D B. XIV If {X;} and tAj) are the eigenvalues of A and B respectively, then (Xi + pf} are the eigenvalues of A O B.

Proof Let x and y be the eigenvectors corresponding to the eigenvalues X and p of A and B respectively, then

(A(DB)(x®y) _ (A0I)(x0y)+(10B)(x(3y) = (Ax ®y) + (x ®By) = X(x ®y) + U(x ®y)

_ (X+p)(x®y) The result follows.

by (2.19) by (2.11)

Definition of the Kronecker Sum

Sec. 2.41

31

Example 2. S

Verify the Property XN for A

_

-1

l

I

and

B =

0

1-0 1 Cl-lJ

Solution For the matrix A; X, = 1

and

x, = 1101

X2 = 2 and x2 =

-

'=9

For the matrix B;

µ, = 1

[1

and

1122 and

Yi

1i

Y2=L1

We find 2

0 -1 0 0 0 -1

0

0

3

0

0

0

2

1

2

C=AO+B =

(L

and 1 pi - Cl = p (p - 1) (p - 2) (p - 3), so that the eigenvalues of A O B are

and

p = 0 = X, + µ2

and

xt O y2 = [0

1

p = 1 = X2 + 112

and

x2 O Y2 = 10

1 0 -1]'

p = 2 = X,+µr

and

x1Oy, = (1

1

p = 3 = X2 + µr

and

x2 O Yr = 11

1 -1 -1 ]'

0 0]' 0 0]' .

The Kronecker sum frequently turns up when we are considering equations of the form; (2.20) AX + XB = C c,.

where A(n X n), B(m X in) and X(n X m). Use (2.13) and solution to Example 2.3 to write the above in the form vecC or

(11' (D A) vec X = vec C

It is interesting to note the generality of the Kronecker sum. For example,

exp (A + B) = exp A exp B

(2.21)

[Ch. 2

The Kronecker Product

32

if and only if A and B commute (see [ 181 p. 227) 't7

"t7

whereas exp (A 0 B) = exp (A 0 1) exp (I 0 B) even if A and B do not commute! Example 2.6 Show that

exp (A ®B) = expA © exp B

whereA(n X it),B(m X m). Solution By (2.11)

A®B and (A

0 Im) and (In 0 B) commute so that

exp (A ®B) = exp (A 01m + In 0 B) = exp (A ®I,,,) exp (In ®B) = (expA ®Im) (1 ® exp B)

(by 2.15 and 2.16)

= expA 0 expB

(by 2.11)

2.5 THE PERMUTATION MATRIX ASSOCIATING vec X AND vec X'

If X = [x;l] is a matrix of order (in X n) we can write (see (1.20))

X = EEx,/E;j where Eli is an elementary matrix of order (in X n). It follows that

X' = so that

vec X' = EEx11 vec Erl'

(2.22)

.

We can write (2.22) in a form of matrix multiplication as x11

...

x21

.. vec E,;1 vec E12:... vec E,;,n]

I

x,,,, x12 ,,,

vec X' = [vec E11 vec E21

xmn

Sec. 2.5]

The Permutation Matrix

33

that is

vec X' = [vec E11 vec E21; ... vec E,,',,: vec E12 ... vec E,,',,j vec X. So the permutation matrix associating vec X and vec X' is U = [vec E,',

(2.23)

... vec

vec E2,

Example 2.7 Given

X = xli X12 X13

determine the matrix U

x21 x21 x13

such that vecX' = U vec X, Solution

EI'1

0l 1

00

Ei'r

!f

00

El, ' =

1

0

0 0

r f=7

0

0

1,0

E22 =

0

1

0 0

-,

0 0

E23 =

and

0

E13

00

00

= r0

0

1(0)

1

Hence by (2.23) 1

0 0 0 0 0

0 0

1

0 0 0

0 0 0 0 0

--»

U =

1

1

0

0 0 0 0

0 0 0

1

0 0

0 0 0 0 0

1

We now obtain the permutation matrix U in a useful form as a Kronecker product of elementry matrices. As it is necessary to be precise about the suffixes of the elementary matrices, we will use the notation explained at the end of Chapter 1. As above, we write m

X' = > > xrsEsr (n X m)

.

r=l s=1

By (1.31) we can write

X'

Er (nXm)XEsr(11 Xm). r, s

[Ch. 2

The Kronecker Product

34

1 fence,

vec X' = vec Esr (n X nt) XE,rr (n X m) r, s

Er,.(mXn)©E,.r(nXm)jvecX

by (2.13)

r' s

It follows that U = ) Ers (m X n) O Esr (n X m)

(2.24)

r, s

or in our less rigorous notation

U = ,E, Ox Ers

(2.25)

r, s

r..

Notice that U is a matrix of order (nut X nut). At first sight it may appear that the evaluation of the permutation matrices Ut and U2 in (2.14) using (2.24) is a major task. In fact this is one of the examples where the practice is much easier than the theory. We can readily determine the form of a permutation matrix - as in Example 2.7. So the only real problem is to determine the orders of the two matrices. c..

4U.

Since the matrices forming the product (2.14) must be conformable, the orders of the matrices Ut and U2 are determined respectively by the number of rows and the number of columns of (A O B). Example 2.8

Let A = [a111 be a matrix of order (2 X 3), and B = [bit] be a matrix of order (2 X 2). Determine the permutation matrices Ut and U2 such that

A O B = Ut (B 0 A) U2 Solution

(A ©B) is of the order (4 X 6)

From the above discussion we conclude that Ut is of order (4 X 4) and U2 is of order (6 X 6). 0 0 0 0 0 1

1

0 0 0

Ut _ 0 0 0

1

1

1

1

0 0 0

0 0 0 0

0

00

000

0 0 and

U2 =

1

0

0000 00 000 0

1

1

0 0 0 0 0 11

The Permutation Matrix

Sec. 2.51

35

Another related matrix which will be used (in Chapter 6) is

U=

rs O

Ers

(2.26)

r, s

When the matrix X is or order (in X n), U is or order (nr2 X n2).

Problems of Chapter 2

(1) Given

Ers(inX n)0Esr(nX m).

U= r, s

Show that

U-1 = U' =.Er(nXin)0Ers(inXn) r' s C)'

direct method to evaluate

(a) (i) AYB (ii) B' ©A (b) Verify (2.13) that vecAYB = (B' O A) vec Y.

(3) Given A =

r2

-1

1

1

and B = 0

2

0

1

(a) Calculate

AOB add BOA. (-)

(b) Find matrices U, and U2 such that

AOB = Ul(BOA)U2. (4) Given C3

A 2

4

_3

calculate

(a) exp (A)

(b)'exp(A 01). Verify (2.16), that is

exp (A) 01 = exp (A 01).

`L7

(2) A = [at1], B = [b,1] and Y = [y,j] are matrices all of order (2 X 2), use a

[Ch. 2)

The Kronecker Product

36 (5) Given

2

A

-1

1

-

and B

1

1

2

3

4

,

calculate

../

(a) A"' O and B-' (b) (A ©B)'' . Hence verify (2.12), that is

(A © B)'' = A"' © B''

(6) Given

A = L4

2]

and B = L2

,

find

3

(a) The eigenvalues and eigenvectors of A and B. (b) The eigenvalues and eigenvectors of A © B. (c) Verify Property IX of Kronecker Products.

(7) A, B, C and D are matrices such that A is similar to C, and B is similar to D.

Show that A 0 B is similar to C rJ D.

CHAvrER 3

Some Applications of the Kronecker Product

3.1 INTRODUCTION

There are numerous applications of the Kronecker product in various fields including statistics, economics, optimisation and control. It is not our intention to discuss applications in all these fields, just a selected number to give an idea

"'7

..'

.'^

....

,-.

ice'

C].

of the problems tackled in some of the literature mentioned in the Bibliography. There is no doubt that the interested reader will find there various other applications hopefully in his own field of interest. A number of the applications involve the derivative of a matrix - it is a well known concept (for example see [18] p. 229) which we now briefly review. 3.2 THE DERIVATIVE OF A MATRIX Given the matrix

A(t) _ [ar!(t)) the derivative of the matrix, with respect to a scalar variable t, denoted by (d/dt)A(t) or just dA/dt or A(t) is defined as the matrix

dtA(t) - dta;t(t)I

.

(3.1)

I

Similarly, the integral of the matrix is defined as

JA(t)dt = [Jaii@)dt For example, given

A =

2t2

4

sin t

2 + t2

(3.2)

Some Applications of the Kronecker Product

38

[Clt. 3

then

d

Q

14t

fAdt =

and

dt A = cost 2t

t3

4t

-cost 2t + t3/3

+C

where C is a constant matrix. One important property follows immediately. Given conformable matrices A(t) and B(t), then

dt [AB] = aAB+A d-

.

(3.3)

cry

Example 3.1 Given

C = AOB

(each matrix is assumed to be a function of t) show that

717

dC

= dAOB+AO dB

3.4)

Solution On differentiating the (i, j)th block of A O B, we obtain

t d

(aijB)

i'iB + a,i aB

which is the (i, j)th partition of

dAOB+AOdB

,

the result follows.

3.3 PROBLEM 1

Determine the condition for the equation

AX+XB = C to have a unique solution. Solution We have already considered this equation and wrote it (2.21) as

(B'@ A) vec X = vec C or

Gx = c where G = B' (D A and c = vec C.

(3.5)

Problem I

...

Sec. 3.3)

39 "..

Equation (3,5) has a unique solution 1ff G is nonsingular, that is iff the eigenvalues of G are all nonzero. Since, by Property XIV (see section 2.4), the eigenvalues of G are (X1 + µ/) (note that the eigenvalues of the matrix B' are the same as the eigenvalues of B). Equation (3.5) has a unique solution iff

Xr+µl a:0

(all iandj).

We have thus proved that AX + BX = C has a unique solution iff A and (-B) have no eigenvalue in common. If on the other hand,A and (-B) have common eigenvalues then the existence of solutions depends on the rank of the augmented matrix

[Gc] If the rank of [G:c] is equal to the rank of G, then solutions do exist, otherwise the set of equations

AX+XB = C is not consistent. Example 3.2 Obtain the solution to

AX+XB = C (1)

(ii)

A=

I0

A=1

22

0

,

[

B= B=

2,

_;'

where

1

3 +

0]

an d

12

4

0 -1

an d

^'r

Solution Writing the equation in the form of (3.5) we obtain, (1)

-2 - 1 0-1

1

0

x1

I

0

1

x2

-2

0

1-1

x3

3

0

4

0

x4

2

where for convenience we have denoted

x2

x,l

-N1

4

2

C=

C=

2

2 -9

Some Applications of the Kronecker Product

40

[Ch. 3

00.

On solving we obtain the unique solution

X=

10

21 -1

1

(ii) In case (ii) A and (-B) have one eigenvalue (X = 1) in common. Equation (3.5) becomes

H2 -1

0

0

x,

0 -1

0

0

x2

4

0

0 -1

x3

S

LO

4

0

x4

-9

J

0

_

2

and rank G = rank [G; c]. G is seen to be singular, but

rank G = rank [G c] = 3

1

...

hence at least one solution exists. In fact two linearly independent solutions are

Ti

0

X, _

and

-2 -1

X2 =

1-1

-2 -1

0

any other solution is a linear combination of X, and X.2.

3.4 PROBLEM 2

Determine the condition for the equation

AX-XA=yX

(3.6)

to have a nontrivial solution.

Solution We can write (3.6) as

Hx = px

(3.7)

whereH=I®A -A'@ I and

x = vecX . (3.7) has a nontrivial solution for x iff

1,41-HI = 0 0

that is iff p is an eigenvalue of H. But by a simple generalisation of Property XIV,

Sec. 3.5]

Problem 3

41

section 2.4, the eigenvalues of H are {(At - ?l)} where {rr} are the eigenvalues of A. 1-fence (3.6) has a nontrivial solution iff

p= Example 3.3 Determine the solutions to (3.6) when

A =

5 of 2

and

p = -2 .

3

Solution

p = -2 is an eigenvalue of H, hence we expect a nontrivial solution. Equation (3.7) becomes 0

0--2

01

0-2

XI

X1

2

2

0

0 -2

0

X3

X3

0

0

0

xa

x4

2

x2

x2

= -2

On solving, we obtain 1

X=

1

-1 -1

3.5 PROBLEM 3

Use the fact (see [18] p. 230) that the solution to is

z = Ax , x(0) = c

(3.8)

x = exp (A t) c

(3.9)

to solve the equation

X = AX + XB

,

X (O) = C

(3.10)

where A(n X n), B(m X in) and X(n X rn). Solution Using the vec operator on (3.10) we obtain where and

X = GX ,

x (0) = c

x = vecX, c = vecC

G = I,,, OA+B'OI

(3.11)

[Ch. 3

Some Applications of the Kronecker Product

42

By (3.9) the solution to (3.11) is

vee X = exp {(I,,, 0 A) t + (B' ®lr,)t) vcc C vcc C (see Example 2.6) [exp (I,,, ©A)t] [exp [I, © exp (At)] [exp (Bt) O

by (2.17) and

vec C

(2.18).

We now make use of the result

vec AB = (B'(D I) vec A

(in (2.13) put A =1 and Y - A) in conjunction with the fact that

[exp (B'r)] = exp (Bt) , to obtain

vec C = vec [Cexp (Bt)]

(exp (Bt) O

Using the result of Example 2.3(1), we finally obtain vec X = vec [exp (At) C exp (Bt)

(3.12)

So that X = exp (At) C exp (Bt). Example 3.4 Obtain the solution to (3.10) when -1 ,

0

B=

0

1

and

-2

C=

0 -1

2

0

.-.

1

A =

1

I

Solution (See [ 18] p. 227)

exp(At) = hence

er

et - e2t

fl

e2t

,

exp (Bt) =

et 0

0

l

er

_e2r-ear

X e3t

er

3.6 PROBLEM 4 We consider a problem similar to the previous one but in a different context. An important concept in Control Theory is the transition matrix. Very briefly, associated with the equations

X = A(t)X or is = A(t)x is the transition matrix (P1 (t, r) having the following two properties

...

c1(t r) = A(t)'t1(t r) and

`b1(t, t) = 1

(3.13)

Problem 4

Sec. 3.6J

[For simplicity of notation we shall write it is easily shown that

43

for cb(t,T).] lfA is a constant matrix,

(1) = exp(At) Similarly, with the equation

X = X13 so that

X' = 13'X'

we associate the transition matrix 4'2 such that (3.14) 4,2 = B'`F2 . The problem is to find the transition matrix associated with the equation

X=AX+XB

(3.15) ,L]

given the transition matrices 4' and `I'2 defined above.

Solution We can write (3.15) as

is=Gx where x and G were defined in the previous problem. We define a matrix as (3.16)

Ji(t,T) _ 1,2(t,T)0 `P,(t,T) We obtain by (3.4) q)2 ©(>;, +

4)2

by (3.13) and (3.14)

(B'4'2) ® `1't + `1'2 O (A`1)1)

= (B'`F2) ® (I`1't) + (I`F2) ® (A`l't)

= [B'OI+IOAi[(2O(1?,J Hence

by (2.11)

.

=GO .

Also

i (t, t) _ `l'2(4 r) ®`F (t, r)

= I®I =I.

(3.18) .ti

The two equations (3.17) and (3.18) prove that L is the transition matrix for (3.15) Example 3.5 Find the transition matrix for the equation

X

-

X+X IO

2

1

0

0 -1

Some Applications of the Kronecker Product

44

(Clr. 3

Solution In this case both A and B are constant matrices. From Example 3.4.

et-e2` i

et

4'1 = exp(At) =

e2t

Lo Iet

4)2 = exp (Bt) _

coy

0 t e-

0

So that

e2t e2t__e31

1G=(D2O t=

0

0

0

0

0

1

1 -et

0

0

et

0

e

0 Lo

3t

For this equation

2 -1

0

0

0

3

0

0

0

0

0 -1

L0

0

0

G =

1,

and it is easily verified that

=Gi and

3.7 PROBLEM 5 Solve the equation

AXB =C where all matrices are of order n X n.

Solution Using (2.13) we can write (3.19) In the form Hx = c

(3.20)

where H = B'O A, x = vec X and c = vec C. The criteria for the existence and the uniqueness of a solution to (3.20) are well known (see for example [ 18] ). The above method of solving the problem is easily generalised to the linear equation of the form A1XB1 + A2XB2 + ... +A,XB,. = C (3.21)

Problem 6

Sec. 3.81

45

Equation (3.21) can be written as for example (3.20) where this time B,

Example 3.6 Find the matrix X, given

A1XB1 +A2XB2 = C where

02 B2 =

-l

4 -6

1

and

3

C= 1

Solution For this example it is found that

H = B0A1+Bz0A2 =

r

0

8

2

2 -2 - 3

--t

1 -1

1

2

0

2

5

2

-4 -2 -5 - 4

andc'=[4 0 -6 81 It follows that -1

x = H-lc =

-2 0

so that

X

3.8 PROBLEM 6

This problem is to determine a constant output feedback matrix K so that the closed loop matrix of a system has preassigned eigenvalues. A multivariable system is defined by the equations

x = Ax+Bu

y=Cx

(3.22)

where A(n X n), B(n X m) and C(r X n) are constant matrices, u, x and y are column vectors of order in, n and r respectively.

Some Applications of the Kronecker Product

[Ch. 3

We are concerned with a system having an output feedback law of the form

u = Ky

(3.23)

where K(m X r) is the constant control matrix to be determined.

On substituting (3.23) into (3.22), we obtain the equations of the closed loop system

z=(A+BKC)x y

= Cx

(3.24)

.

The problem can now be restated as follows: Given the matrices A, B, and C, determine a matrix K such that

XI -A -BKC I = ao + a, X + ... + an._1 A"-1 + A" (say)

(3.25) = 0 for preassigned values A = A1, A2, ..., An

Solution

Various solutions exist to this problem. We are interested in the application of the Kronecker product and will follow a method suggested in [24]. We consider a matrix H(n X n) whose eigenvalues are the desired values A1, A2 ... , An, that is

IAl-HI = 0 and

for

A=

IAl-HI = ao+a1A+...+an_1A"-1+A"

Let

(3.26) .

(3.27)

A + BKC = H

so that

BKC=H-A=Q (say)

3.28)

Using (2.13) we can write (3.28) as

(C'@ B) vec K = vec Q

(3.29)

or more simply as

Pk = q

(3.30)

where P = C' O B, k = vec K and q = vec Q. Notice that P is of order (n2 X mr) and k and q are column vectors of order mr and n2 respectively. The system of equations (3.30) is overdetennined unless of course to = n =r,

in which case can be solved in the usual manner - assuming a solution does exist!

In general, to solve the system for k we must consider the subsystem of linearly independent equations, the ienraining equations being linearly dependent

I!]

Problem 6

Sec. 3,8 .1

47

on this subsystem. In other words we determine a nonsingular matrix T(n2 X n2) such that

Pt

TP = ---

(3.31)

Li P2 .°,

where P, is the matrix of the coefficients of the linearly independent equations of the system (3.30) and P2 is a null matrix.

Premultiplying both sides of (3.30) by T and making use of (3.31), we obtain

TPk=Tq

or

u

k=

(3.32)

V

LiPN

..Y

'.'

oCD

+'7

'r.

If the rank of P is tnr, then Pl is of order (nir X rnr), P2 is of order ([n2- mr] X mr) and u and v are of order nir and (n2 -mr) respectively. A sufficient condition for the existence of a solution to (3.32) or equivalently to (3.30) is that v = 0

(3.33)

k = Pt-t u

may.'

in (3.32). If the condition (3.33) holds and rank Pt = mr, then (3.34)

,

The condition (3.33) depends on an appropriate choice of H. The underlying assumption being made is that a matrix H satisfying this condition does exist. This in turn depends on the system under consideration, for example whether it .°C

is controllable. Some obvious choices for the forth of matrix H are: (a) diagonal, (b) upper or lower triangular, (c) companion form or (d) certain combinations of the above forms. Although forms (a) and (b) are well known, the companion form is less well documented. Very briefly, the matrix .'.

H=

0

1

0

...

0

0

0

I

...

0

0

0

0

...

1

Lao -ar -a2

-a"-

is said to be in `companion' form, it has the associated characteristic equation

IA! -HI = ao + at?t + ... +

0

(3.35)

(Ch. 3

Some Applications of the Kxonecker Product Example 3. 7

Determine the feedback matrix K so that the two input - two output system 0

1

x= 3

3

2 -3

0

0

0

1 x+

1

0u

2

0

1

has closed loop eigenvalues (-1, -2, -3). Solution

We must first decide on the form of the matrix H. Since (see (3.28))

H - A = BKC and the first row of B is zero, it follows that the first row of

H-A must be zero. We must therefore choose H in the companion form.

Since the characteristic equation of His

(X+1)(X+2)(a+3) = X3 +6X2+11a+ 6 = 0 H

1

0

0

1

(see (3 .35))

[-6 -11 --6 ONO

0

and hence (see (3.28)) 0

0

Q = -3 -3

0

r0

-8 -8 -8

1

P=C'OB=

11

110

0

0

0

0

1

0

1

0

0

1

0

1

.

1

O

0

0

0

0

0

0

1

0

1

0

1

0

0

1

0

1

0

1

0

0

0

0

0

0

1

0

0

0

0

1

Problem 6

Sec. 3.8]

49

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

T=

0

An appropriate matrix T is the following

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

1

0 0-1

0

0

0

0

1

0

0-1

0

0

0

0

0

0

0

0

0

0

1

0

It follows that 0

1

0

0

0

1

1

0

1

0

0

1

0

1

P,

0

0

0

0

PZ

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

`--O

0

TP =

and

0 - 8 - 3 - 8

Tq =

0

u v

0

0

Some Applications of the Kronecker Product

50 Since

Pt =

0

0

1

0

0

0

0

1

0

1

0

1

0

1

L0

1J

so that (see (3.34) )

-3

k =Pi-lu =

0

0

-8 Hence

A_

[-13

0

-8

-1

Pit =

0

1

0

0-1

0

1

1

0

0

0

0

1

0

0

[Ch. 3]

CHAPTER 4

Introduction to Matrix Calculus 4.1 INTRODUCTION

.CC

n';

SAC

It is becoming ever increasingly clear that there is a real need for matrix calculus in fields such as multivariate analysis. There is a strong analogy here with matrix algebra which is such a powerful and elegant tool in the study of linear systems and elsewhere. Expressions in multivariate analysis can be written in terms of scalar calculus, fl..

but the compactness of the equivalent relations in terms of matrices not only .ti

leads to a better understanding of the problems involved, but also encourages the consideration of problems which may be too complex to tackle by scalar calculus. We have already defined the derivative of a matrix with respect to a scalar (see (3.1)), we now generalise this concept. The process is frequently referred to v0,

'C7

".,

.w,

II.

as formal or symbolic matrix differentiation. The basic definitions involve the partial differentiation of scalar matrix functions with respect to all the `'"

elements of a matrix. These derivatives are the elements of a matrix, of the same order as the original matrix, which is defined as the derived matrix. The words 'formal' and 'symbolic' refer to the fact that the matrix derivatives are defined without the rigorous mathematical justification which we expect for the corresponding scalar derivatives. This is not to say that such justification cannot be made, rather the fact is that this topic is still in its infancy and that appropriate mathematical basis is being laid as the subject develops. With this in mind we make the following observations about the notation used. In general the elements 't7

.."

coo

-L7

of the matrices A, B, C, . will be constant scalars. On the other hand the elements of the matrices X, Y, Z, . . are scalar variables and we exclude the possibility that any element can be a constant or zero. In general we will also demand that these elements are independent. When this is not the case, for example when the matrix X is symmetric, is considered as a special case. The . .

a,:

.

f-'

try

t17

.-.

,w.

°.'

(G9

reader will appreciate the necessity for these restrictions when he considers the partial derivatives of (say) a matrix X with respect to one of its elements xr5. Obviously the derivative is undefined if xr,. is a constant. The derivative is Er,s if xr5 is independent of all the other elements of X, but is Er,s + E,,. if X is symmetric.

Introduction to Matrix Calculus

52

(Ch. 4

There have been attempts to define the derivative when xrs is a constant (or Zero) but, as far as this author knows, no rigorous mathematical theory for the general case has been proposed and successfully applied.

4.2 THE DERIVATIVES OF VECTORS Let x and y be vectors of orders n and m respectively. We can define various derivatives in the following way (15]:

(1) The derivative of the vector y with respect to vector x is the matrix row

ays

aYR,

ax,

ax,

ax,

ay

ayt

rod`

INS

FaYt

ax

ax2

ax2

aYm

3Y2

((d

axe

ay,

by.,

aym

axn

axn

ax-1

(4.1)

of order (n X m) where yr, Y2, ... , y,,, and x,, x2, ... , x are the components of y and x respectively. C1'

`d.

(2) The derivatives of a scalar with respect to a vector. Ify is a scalar ray

ay

T

ax

ax2

ay

(4.2)

by axn

(3) The derivative of a vector y with respect to a scalar x ay,

loo

aye

ax

Lax

ax

Example 4.1 Given

y=

rpm

by

aym

ax X,

Yr

x=

x2

Y2 X3

(4.3)

Sec. 4.2]

53

The Derivatives of Vectors

and

Yi =xi-x2 Y2 = x3 + 3x2

Obtain ay/ax.

Solution

ay

_

3Yi

ay-2

axe

axt

ay,

aye

axe

axe

ay,

ay2

ax3

ax3

2xt\ 0 -1

ax

0

3

2xj

In multivariate analysis, if x and y are of the same order, the absolute value of the determinant of ax/ay, that is of aX

ayJ is called the Jacobian of the transformation determined by

y = Y(x) .ti

Example 4.2

/Ay

The transformation from spherical to Cartesian co-ordinates is defined by x = r sin 0 cos >V ,y = r sin B sin ', and z = r cos B where r > 0, 0 < 0 . A=l

/',. ail bpjxlp

(4,14)

.

1=1

From (4.14) we immediately obtain

roles

ayij axrs

(4.15)

- atrbsj

We can now write the expression for aylj/aX , ayii

ayij

aytj

ax11

aX12

a

ayu

ayu

ax21

ax22

aX2n

, .

aXln

(4.16)

... aylj

ay11

aXm 1 axm 2

...

aye,

axm n

Using (4.15), we obtain

ailbnj

a12b2j

... ...

Limblj aimb2i

...

almbnjj

a11blj

aitb2j

ai2blj

aylj

ax

a12bnj

(4.17)

We note that the matrix on the right hand side of (4.17) can be expressed as (for notation see (1.5) (1.13) (1.16) and (1.17)) ail

ail

(btjb2j ... bnjj

atmj

= Al. B./ = A'e1 ee B'.

(Ch. 4

Introduction to Matrix Calculus

62

So that ay`I

ax

= A'Er/B'

(4.18)

where Ell is an elementary matrix of order (I X q) the order of the matrix Y. We also use (4.14) to obtain an expression for aYlaxrs

=

aXrs

r0'

aY

(r, s fixed, 1, j variable I < i s 1, 1< j 5 q)

ay,I aXrs

that is ayl2

ay,g

axrs

aXrs

aXrs

ay

aye,

ay22

ay2g

axrs

aXrs

axrs

axrs

ay a

8YI2

xs

axrs

Eli ay" axrs

(4.19)

AID"

ayll

aytq axrs

where Et1 is an elementary matrix of order (1 X q). We again use (4.15) to write

ayu

alrbsl

alrbs2

a2rbsi

a2rbs2

arnrbsl

arnrba2

... ... alrbq a2rbsq

axrs .

amrbsq

air

a2r

[bst b52

.

. .

bsq ]

arnr

A.rBs' = AeresB

.

So that axrs

= AErsB SRS

a (AXB)

(4.20)

where Ers is an elementary matrix of order (m X n), the order of the matrix X.

The Derivative of a Matrix

Sec. 4.51

63

Example 4.5 Find the derivative aY/axr,, given

Y = AX'B where the order of the matrices A, X and B is such that the product on the right hand side is defined. Solution By the method used above to obtain the derivative a/axis (AYB), we find a 3Xrs

(AX'B) = AE,,B

.

Before continuing with further examples we need a rule for determining the derivative of a product of matrices. Consider

Y = UV

(4.21) .C)

where U = [u11] is of order (rn X n) and V = [qj] is of order (n X 1) and both U and V are functions of a matrix X. We wish to determine

aY

ay11

axis

ax

-- and

The (i,j)th element of (4.21) is n

ylj =

(4.22)

UIPVPI

P=1

hence

airs

n

vPj P=

i aXrs

P=I

-

avP1 U iP

lam

n UUp

ay;j

(4 23)

.

.

axis

For fixed r and s, (4.23) is the (i,j)th element of the matrix aYlax,s of order (m X 1) the same as the order of the matrix Y. II.

On comparing both the terms on the right hand side of (4.23) with (4.22), we can write

a(UV) axrs

as one would expect.

au axis

V+U

av axis

(4.24)

Introduction to Matrix Calculus

64

[Ch. 4 ti-

On the other hand, when fixing (i,j), (4.23) is the (r,s)th element of the matrix ay;l/aX, which is of the same order as the matrix X, that is ay,l

L "lip ax vpl + L utp

ax

p=1

p-1

avpl (4.25)

ax

We will make use of the result (4.24) in some of the subsequent examples. Example 4.6 Let X = [xrs] be a non-singular matrix. Find the derivative aY/axrs, given

(i) Y = AX -'B, and

(ii) Y=XAX Solution

(i) Using (4.24) to differentiate

yy-t = I, we obtain

aY 3Y-' = 0, -Y-'+Y axrs axrs

hence

aY -axrs _ -Y -ay-' Y-. axrs

But by (4.20) a

axrs

axrs

(B-1XA-1) = B-'Ers q-t

,-.

3Y-' so that

CID

ay

a

- = - (AX-'B) = AX -'BB-'ErsA-'AX -'B axrs

axrs

AX-'ErsX-'B .

(ii) Using (4.24), we obtain

ay axrs

_

-

aX'

axrs

AX+X'

a(AX) axrs

_ E, AX + X'Airrs

(by (4.12) and (4.20)) .

Both (4.18) and (4.20) were derived from (4.15) which is valid for all i, j and r, s, defined by the orders of the matrices involved. 1

The Derivative of a Matrix

Sec. 4.5 1

65

The First Transformation Principle

,R.

It follows that (4.18) is a transformation of (4.20) and conversely. To obtain (4.18) from (4.20) we replace A by A', B by B' and Er: by Eli (careful, Ers and Etl may be of different orders). The point is that although (4.18) and (4.20) were derived for

constant matrices A and B, the above transformation is independent of the status of the matrices and is valid even when A and B are functions of X.

Example 4.7 Find the derivative of aytl/aX, given

(i) Y = AX'B,

(ii) Y=AX-'B, and (iii) Y = X AU where X = [x,l] is a nonsingular matrix, Solution (1) Let W = X', tlien

ay Y = AWB so that by (4.20) - =AEr3B aWrs

hence

ay,l

aw

= A'E;iB'.

But ayL/

a}ri

ax

aw'

_ (ay.l

awl

hence DYq

ax

= BE ;IA

(ii) From Example 4.6(i) aY axrs

-AX-'L,-,,,X-'B.

Let At = AX -1 and Bt = X''B, then aY a xrs

A1E 3B

1

so that ay,t

ax

= AiE,1B1' = -(X )'A'E;1B'(X t)' .

Introduction to Matrix Calculus

66

[Ch. 4

(iii) From Example 4.6 (ii) aY aXrs

= E,,AX + X'AE,,s

.

0.j

LetA,=1,Bt=Ax,A2=XAandB2=1, then ax axrs

= AtErsBl +A2Ersl32 .

The second term on the right hand side is in standard form. The first term is in the form of the solution to Example 4.5 for which the derivative ay;l/aX was found in (i) above, hence ay 'r = B1E;1AI + A2E,/B2'

ax

= AXE; + A`xE;l . It is interesting to compare this last result with the example in section 4.2 when we considered the scalary = x'Ax. In this special case when the matrix X has only one column, the elementary matrix which is of the same order as Y, becomes

E;1=E;j=1. Hence

ay,, = aY

ax

ax

= Ax + A'x

which is the result obtained in section 4.2 (see (4,4)). Conversely using the above techniques we can also obtain the derivatives of the matrix equivalents of the other equations in the table (4.4). Example 4.8 Find aY

ay;; and

aXrs

ax

when (i) Y = AX, and

(ii) Y=X'X. Solution (i) With B = I, apply (4.20) aY axrs

= AEr3.

The Derivatives of the Powers of a Matrix

Sec. 4.61

The transformation principle results in ay11

ax

(ii) This is a special case of Example 4.6 (ii) in which A = I. We have found the solution aY axrs

ErsX + X'Ers

and (Solution to Example 4.7 (iii))

'Y" = XE11 + XEj . ax

4.6 THE DERIVATIVES OF THE POWERS OF A MATRIX Our aim in this section is to obtain the rules for determining ay;;

ay and

axrs

when

ax

Y=X".

Using (4.24) when U = V= X so that

Y=X2 we immediately obtain

ay

- =ErsX+XErs axrs

and, applying the first transformation principle,

ay,

ax

= E;1X'+X'E;j .

It is instructive to repeat this exercise with

U= X 2 so that Y

and

V= X

X3.

We obtain

ay Ltd

axrs

= ErsX 2 + XErsX + X 2Ers

and

Y-u = Ei, (X')2 + X'EifX' + (X 1)2E,,

ax

67

Introduction to, Matrix Calculus

68

[Ch. 4

More generally, it can be proved by induction, that for

Y=Xn kEESXn-k-1

X

(4.26)

k=0

where by definition X ° = I, and

"-I

ay;l

(4.27)

x )k E,j (X ) n -k-1

ax

k=1

Example 4.9 Using the result (4.26), obtain aYlaxrs when

Y=X-n Solution Using (4.24) on both sides of

X-nXn=I we find

a(X-n)

Xn+X-n

a (Xn)

airs

so that

=

0

axrs

3(X-n)

_ `x -n a(Xn) X-n.

axrs

axrs

Now making use of (4.26), we conclude that

3(X-n)

= -x-n

7

Fn-1

XkErsXn-k-1

axrs

L=° Problems for Chapter 4

...

-

(1) Given

x=

xtl x12 x3 x233]

Y=

x21 x22

and y = 2x11x22 -x21x13, calculate BY ay

ax

and

ax

x-1 1

2x2

sin x

The Derivatives of the Power of a Matrix

Sec. 4.61

69

(2) Given

X

sinx

X

cos x

cz

X=

and

Fsinx

ex

L'

XI

evaluate

alxl ax by

(a) a direct method (b) use of a derivative formula.

(3) Given X =

X11

x12 X13

and

Y = X'X,

Lx 21 x22 X231

use a direct method to evaluate (a)

DY

and

(b)

aY i3

ax-21

ax

(4) Obtain expressions for

by ax's

and

ay;;

ax

when

(a) Y = XAX and

(b) Y = XAX'.

(5) Obtain an expression for atAXBI/ax,,. It is assumedAXB is non-singular. (6) Evaluate aY/ax,,s when (a) Y = X (X')2

and

(b) Y = (X')2X.

CHAPTER 5

Further Development of Matrix Calculus including an Application

of Kronecker Products 5.1 INTRODUCTION

In Chapter 4 we discussed rules for determining the derivatives of a vector and then the derivatives of a matrix. But it will be remembered that when Y is a matrix, then vec Y is a vector.

This fact, together with the closely related Kronecker product techniques discussed in Chapter 2 will now be exploited to derive some interesting results. Also we explore further the derivatives of some scalar functions with respect to a matrix first considered in the previous chapter. 5.2 DERIVATIVES OF MATRICES AND KRONECKER PRODUCTS In the previous chapter we have found ay;!/3X when

Y = AXB

(5.1)

-o^

where Y = [y1j], A = [ajj], X = [x11] and B = [by]. We now obtain (a vec Y)/(a vec X) for (5.1). We can write (5.1) as

y=Px

(5.2)

where y = vec Y, x=vecXand P=B'OA. By (4.1), (4.4) and (2.10) ay

ax

=P' = (B'OA)' = BOA'.

(5.3)

The corresponding result for the equation

Y = AX'B is not so simple.

(5.4)

[Sec. 5.2]

time

Derivatives of Matrices and Kronecker Products

71

The problem is that when we write (5.4) in the form of (5.2), we have this

y = Pz

(5.5)

where z = vec X' We can find (see (2.25)) a permutation matrix U such that

vecX' = UvecX

(5.6)

in which case (5.5) becomes

y=PUx so that ax

= (PU)' = U'(B ®A') .

5.7)

It is convenient to write

U'(B O A') = (B

(5.8)

U' is seen to premultiply the matrix (B O A'). Its effect is therefore to rearrange the rows of (B d A'). In fact the first and every subsequent nth row of (B (D A') form the first consecutive m rows of (B O A')(,,). The second and every subsequent nth row form the next m consecutive rows of (B and so on. A special case of this notation is for n = 1, then

(B (D A'){1) = BOA'

.

(S.9)

Now, returning to (5.5), we obtain, by comparison with (5.3) ay ax

= (B(D

Example 5.1 Obtain (a vec Y)/(a vec X), given X = [x;l] of order (m X n), when

(i) Y=AX, (ii) Y=XA, (iii) Y=AX' and (iv) Y=XA. Solution

Let y = vec Y and x = vec X.

(i) Use (5.3) with B = I ay ax

= 10 A'.

(5.10)

Further Development of Matrix Calculus

72

(ii) Use (5.3) ay

ax

= A ®I .

(iii) Use (5.10) ay

_ (I ®A')(n)

ax

(iv) Use (5.10) ay

= (A ®I)(o

ax

5.3 THE DETERMINATION OF (a vec X)/(3 vec Y) FOR MORE COMPLICATED EQUATIONS

In this section we wish to determine the derivative (a vec Y)/(a vec X) when, for example, Y = X'AX (5.11) wheie X is of order (m X n).

Since Y is a matrix of order (n X n), it follows that vec Y and vec X are vectors of order nn and nm respectively. With the usual notation III

Y = [yi/)

,

X = [xi/)

we have, by definition (4.1), ay11

ay21

ax11

ax11

ax11

a vec Y

ayl I

ay21

aynn

avecx

axle

a .x21

ax21

aynn

ayll

ay21

aynn

aXmn

axmn

3Xmn

...

...

...

...

I

[Ch. 5

But by definition (4.19), ay) ' the first row of the matrix (5,12) is vec -ax, I

/

the second row of the matrix (5.12) is +\vec

a'

Y-),etc.

a.x21

(5.12)

The Determination of (3 vecX)/(3 vec Y)

Sec. 5.3]

73

We can therefore write (5.12) as a vec Y

a vecX

( by aY 1 ' BY = vec - : vec - ; ... ; vec ,

3x11

ax,nn

8x21

(5.13)

We now use the solution to Example (4.6) where we had established that

when Y = X'AX, then

by axrs

= E,,SAX + X AErs .

(5.14)

It follows that

by

vec - = vec E;SAX +vec X AE,S axrs

= (XA'OI) vecE;S+(IOXA)vecErs

(5.15)

(using (2.13)) . Substituting (5.15) into (5.13) we obtain a vec Y

a vec X

_ [(X'A'01)[vee/'1 vecE21;

.

;vecErnr,]]'

+ [(IOXA)[vecEll: vecE21:... vecE,,,n]]' _ [vec Eii: vec E21; ... ; vec E;,,n]'(AX 01) + [vec E11 vec E21 vec E,nn ]' (I (DA'X)

(5.16)

(by (2.10)). The matrix [vec E, 1 , vec E21

.. .

vec Ernn ]

is the unit matrix I of order (mn X mn). Using (2.23) we can write (5.16) as

3vecY

avecX

= U'(AX 01) + (10 A'X) .

That is a vec Y

(5.17)

a vcc X

In the above calculations we have used the derivative a Y/axrs to obtain (3 vec Y)/

(a vecX).

Further Development of Matrix Calculus

74

[Ch. 5

The Second Transformation Principle'-j

Only slight modifications are needed to generalise the above calculations and show that whenever ay

= AErsB + CE,, D

aXrs

where A, B, C and D may be functions of X, then a vec Y cow

avecX

=

(5.18)

We will refer to the above result as the second transformation principle. Example .f.2 Find

avecY

when

avecX

(i) Y = X'X

(ii) Y = AX-'B

Solution

Lety=vecYandx=vecX (i) From Example 4.8 ay

= Er'sX + X'Ers

aXrs

Now use the second transformation principle, to obtain ay

ax

= I©X+(X(D

(u) From Example 4.6 ay axrs

AX-'ErjX-'B

hence

ay

ax

= -(X -'B) O (X-')'A'.

Hopefully, using the above results for matrices, we should be able to rediscover results for the derivatives of vectors considered in Chapter 4.

Sec. 5.4]

More on Derivatives of Scalar Functions

75

For example let X be a column vector x then

y=

Y = X'X becomes

x 'x

(y is a scalar) .

The above result for ay/ax becomes av

= (I0 x)+(x0 1)(1)

ax

0

c..

But the unit vectors involved are of order (n X 1) which, for the one column vector X is (1 X 1). ilence ay

= l ©x + x ©1 ax

(use (5,9))

=x+x=2x

which is the result found in (4.4). 5.4 MORE ON DERIVATIVES OF SCALAR FUNCTIONS WITH RESPECT TO A MATRIX

In section 4.4 we derived a formula, (4.10), which is useful when evaluating 31Y)/3X for a large class of scalar matrix functions defined by Y. .ti

Example.5.3 Evaluate the derivatives a log IX

()

ax

and

aIXIr

(ii)

ax

Solution (i) We have

(log IXD = X t-0

axrs

I I

I

.

axa rs

From Example 4.4,

Hence

alxl Ixl(x-') ax = a log IXI

ax (ii)

alxlr aXrs

_ = (X

= rjXj r-1

(non-symmetric case) .

1) .

a1xl aXrs

Further Development of Matrix Calculus

76

[Ch. 5

Hence

alxlr -- rlXIr(X-1)' ax Traces of matrices form an important class of scalar matrix functions covering a wide range of applications, particularly in statistics in the formulation of least squares and various optimisation problems. Having discussed the evaluation of the derivative a Y/axrs for various products of matrices, we can now apply these results to the evaluation of the derivative

a(tr Y)

ax We first note that

a(tr Y) _ [a(tr Y)1 axrs

(5.19) c^,

ax

JI

where the bracket on the right hand side of (5.19) denotes, (as usual) a matrix of the same order as X, defined by its (r,s)th element. As a consequence of (5.19) or perhaps more clearly seen from the definition (4.7), we note that on transposing X, we have

a(tr Y) '

a(tr Y)

ax

ax'

(5.20) -

Another, and possibly an obvious property of a trace is found when considering the definition of aY/axrs (see (4.19)). Assuming that Y = [yij] is of order (n X n)

tray

=

axrs

ayri+aY22+...+aYnn axrs

3Xrs

axrs

a

- (YI1 + Y22 + .

ay

a (tr Y)

axrs

axrs

tr

Example 5.4 Evaluate

+ Ynn)

axrs

Hence,

a tr(AX) ax

(5.21)

Sec. 5.4]

More on Derivatives of Scalar Functions

77

Solution

a tr(AX) aXrs

= tr

a(AX) by (5.21)

airs

= tr (AE,,)

by Example (4.8)

= tr(E,,A')

since tr Y = tr Y'

= (vec E,.,)' (vec A') by Example (1.4). Hence,

atr(AX) ax

,

= A

As we found in the previous chapter we can use the derivative of the trace of one product to obtain the derivative of the trace of a different product. Example 5.5 Evaluate

a tr (AX')

ax Solution From the previous result a t r (BX)

_ a t r (X'B') = B,

ax

ax

.-1

Let A' = B in the above equation, it follows that

atr(X'A) ax

_

atr(A'X) = A.

ax

The derivatives of traces of more complicated matrix products can be found similarly.

Example 5.6 Evaluate

when

8 (tr Y)

aY

(i) Y = XAX (ii) Y = X AXB Solution It is obvious that (i) follows from (ii) when B = I.

Further Development of Matrix Calculus

78

[Ch. 5

(ii) Y = X1B where X1= X AU.

ay _ axt airs

B

ax-".'

= E,s AXB + X'AEB

(by Example 4.6)

Hence,

tr(aY\ = tr(E,3AXB)+tr(X`AErsB) axrs!)

tr (E,,4AXB) + tr (E,,.4 XB')

= (vec EE,.)' vec (AXB) + (vec Ers)' vec (AXB') .

It follows that

a(trY)

= AXB + A'XB'.

ax

(i) Let B = I in the above equation, we obtain

a(tr Y)

ax

= AX+A'X = (A+A')X .

5.5 THE MATRIX DIFFERENTIAL For a scalar function f(x) where x = [x1 x2 as

df = > J=

of

... x,,]', the differential df is defined

dxl.

(5.23)

Ox,

Corresponding to this definition we define the matrix differential dX for the matrix X = [x;1] of order (m X n) to be

>'C

dX =

dxtn

dx22

... ...

dXm2

...

dxrn.1

dx11

dx12

dx21

Ldxmt

(5.24)

dx2n

.

The following two results follow immediately:

d(aX) = a(dX)

(where a is a scalar)

d(X + Y) = dX + dY. Consider now X = [x;1] of order (m X n) and Y = [ y,f] of order (n X p).

XY = [ExjJyjk]

(5.25)

(5.26)

The Matrix Differential

Sec. 5.5]

79

hence

d(XY) = d[Yxtlyjk) = 7

_ [E(dXij)yjk) + IExii(dYjk)) It follows that

d(XY) = (dX)Y+X(dY).

(5.27)

Example 5.7 Given X = [xtl] a nonsingular matrix, evaluate

(i) dlXl , (il) d(X'') Solution

(i) By (5.23) dIXI

(dx,j) ax11

Xij(dxij) since (a1Xl)/(axij) =Xij, the cofactor ofxij in IXI. By an argument similar to the one used in section 4.4, we can write

dIXI = tr {Z'(dX)}

(compare with (4.10))

where Z = IXij] Since Z'= IX jX-1, we can write

dIXI = IXl tr {X-'(dX)} . (ii) Since

X-1X = we use (5.27) to write

d(X-')X + X-'(dX) = 0. Hence

d(X-') = -X-'(dX)X"' (compare with Example 4.6). Notice that if X is a symmetric matrix, then

x=x' and

(dX)' = dX

.

(5.28)

Further Development of Matrix Calculus

80

[Ch. 5]

Problems for Chapter 5

(1) Consider

A =

all a12 a21

X=

a12

[X11 xiz

and Y = AX'.

X21 X22

Use a direct method to evaluate a vec Y

avac X and verify (5.10).

(2) Obtain avac Y

avecx when

(i) Y = AX'B and (ii) Y = )JAII X2. (3) Find expressions for

atrY ax when .,.,.

(a) Y = AXB, (b) Y = X2

and

(c) Y = XX'.

(4) Evaluate

a try ax when

(a) Y = X-1, (b) Y = AX-'B, (c) Y = X" and (d) Y = eX. (5) (a) Use the direct method to obtain expressions for the matrix differential dY when

(i) Y = AX, (ii) Y = X'X and (iii) Y = X2. (b) Find dY when

Y = AXBX.

Cl IAPTLR 6

The Derivative of a Matrix with respect to a Matrix 6.1 INTRODUCTION

In the previous two chapters we have defined the derivative of a matrix with respect to a scalar and the derivative of a scalar with respect to a matrix. We will now generalise the definitions to include the derivative of a matrix with respect y,,

to a matrix. The author dial"adopted the definition suggested by Vetter [31], although other definitions also'give rise to some useful results.

6.2 THE DEFINITIONS AND SOME RESULTS

Let Y = [y,j be a matrix of order (p X q). We have defined (see (4.19)) the derivative of Y with respect to a scalar xrs, it is the matrix [ayti/axr,s] of order

(pXq) Let X = [xrs] be a matrix of order (m X n) we generalise (4.19) and define the derivative of Y with respect to X, denoted by aY

ax as the partitioned matrix whose (r,s)th partition is aY axrs

in other words ay

ay

OXt1

3x12

aY

aY

aY

ax

421

...

axij aY

...

}d{

OXmt

Cc)

aY

aY

axm2

ay

Ers0 -

_ 3x2n

...

a.X22

aY

aY

r, s

axrs

(6.1)

[Clt. 6

The Derivative of a Matrix with Respect to a Matrix

82

The right hand side of (6.1) following from the definitions (1.4) and (2.1) where Err is of order (in X n), the order of the matrix X. It is seen that 3Y/3X is a matrix of order (mp X nq). Example 6.1 Consider Y =

x11 x12 x22

sin(xii +x12)

exll x" log (x1t ,F-X21))J

and

X

x11 xt21 x21 x22

_.y

Evaluate aY

ax Solution

ay

x22 exl l x]] 1

12 x22

axi t

cos (XI I

1

+ x12)

(x11 + x21)

ay aX12

x77 x22

0

cos (x11 + x12)

0

1

421

0

,1y

0

0

ay

ay ax22

x11x12

x17 exllx731

0

0

x11 + x21 x12 x22

ay ax

x22 exl l x»

X1 t x22

0

cos (x11 + x12)

0

xtt x12

x11 exl l x21

1

cos (x11 + x 12 )

xii + x21

0

0 1

0

0

Example 6.2 Given the matrix X = [xv] of order (m X n), evaluate aX/aX when

(i) All elements of X are independent (ii) X is a symmetric matrix (of course in this case m = n).

0

The Definitions and Some Results

Sec. 6.2)

.-,

Solution (i) I3y (G.1)

ax r

ax

= U (see (2.26))

r, s

ax

= Ers +Esr

axrs

ax

=

axrs.

"

for

r$s

for

r=s

We can write the above as;

ax = Ers + Esr - SrsErr

axrs

Hence, ax

Ers + > Ers Ox Esr ` 5rs > Esr Ox Err

rs

ax

r,s

r,s

r, s

= U+ U-2:ErrOx Err

(see (2.24) and (2.26))

Example 6.3 Evaluate and write out in full ax'lax given X12 X13

X11

X =

Lx21 x22 x231 v°,

.-,

Solution By (6.1) we have ax'

ax = Ers © Ers = U. Hence 1

I--

0

0

0

0

0

0

0

1

0

0

0

ax,

0

0

0

0

1

0

ax -

0

1

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

83

The Derivative of a Matrix with Respect to a Matrix

84

[Ch. 6

From the definition (6.1) we obtain

)'

tax, =(>Ers °aX r, s

by (2. 10)

Ers Ox f a

_

\axr.

O

a Y'

from (4 19)

r,s

It follows that

aY fax

aY (6.2)

= ax'

"C7

'6.3 PRODUCT RULES FOR MATRICES

We shall first obtain a rule for the derivative of a product of matrices with

I-,

respect to a matrix, that is to find an expression for a (XY)

az where the order of the matrices are as indicated

X(mXn), Y(nXv), Z(pXq). By (4.24) we write

a(XY) azrs

=

ax

Y+X

azrs

aY azrs

where Z = [Zrsl

If Ers is an elementary matrix of order (p X q), we make use of (6.1) to write Fax

a (XY)

ay l

Y+X

aZrs

r. s

azrs

ax -Y+

IEr,

aZrs

r, s

Ers(8X

aY azrs

rs

3Y

ax

= > Erslo OX -Y+ r5' IIErs 0X UZrs .ox

aZ

Ers O

r. s

S

UZrs

Product Rules for Matrices

Sec. 6.3 1

85

(where Iq and Ip are unit matrices of order (q X q) and (p X p) respectively)

ax

(Lrs

(D- ) (Iq ®Y) + airs

r, s

(I ®X) Er, rs

aY ---) azrf

(by 2.11)

finally, by (6.1)

a(XY) az

ax = az

(I ®Y) + (I®(@ X) aY

(6.3)

az

Example 6.4 Find an expression for

ax-' ax Solution Using (6.3) on

xX-'=1, we obtain

a (xx-')

ax

ax

ax

ax-1

ax

hence

ax-I ax =

-(I©x)-'

ax(I©x-')

= -(IOX-1)CI(I(& X-') (by Example 6.2 and (2.12)).

Next we determine a rule for the derivative of a Kronecker product of matrices with respect to a matrix, that is an expression for a(X (D Y)

az

The order of the matrix Y is not now restricted, we will consider that it is (u X v). On representing X © Y by it (i,j)th partition [x;1Y] (i = 1, 2, ... , m, k = 1, 2, ..

,

n), we can write

a (X ©Y) azrs

a

air:

[xr1Y]

86

The Derivative of a Matrix with Respect to a Matrix

[Ch. 6

where (r, s) are fixed = L3ZrsYJ + L

aZ

s

j

ax

ay _ aZrs -OY+XO. aZrs Hence by (6.1)

3(X(D Y)

ax

:rs0x -OO Y+

az

aZrs

r, s

r,s

aY E 0X0 aZrs

where Ers is of order (p X q) =aZ®Y+'

Ers0(XO

aY\ azrs J

r,

The summation on the right hand side is not X © aY/aZ as may appear at first sight, nevertheless it can be put into a more convenient form, as a product of matrices. To achieve this aim we make repeated use of (2.8) and (2.11)

Ers®(Xazrs ® aYl= [IpErsIq]OLUii r, s

//

r, s

®X)U1] aZrs

/ by (2.14)

c(0

aY Ers) O U, -0 azrs

r, s

X

[Iq O U2]

by (2.11)

//

ErsOa-Y

OUi]

OO X [Ig0 U2] bY(2.11). aZrs

a(XOY)_ ax0Y+ az

10U ay0X] [IO U21 q

[p

(6.4)

] laz

C1.

az

where U, and U2 are permutation matrices of orders (mu X mu) and (nv X nv) f1.

re pe ctive ly.

We illustrate the use of equation (6.4) with a simple example.

(i) Equation (6.4), and (ii) a direct method to evaluate

a(A©X) ax

c14

'GO

Example 6.5 A = [ail] and X = [x11] are matrices, each of order (2 X 2). Use

Sec. 6.3]

Product Rules for Matrices

87

Solution (i) In this example (6.4) becomes

(Aaxx)

_ [I O

U1 ]

Cax ©A [I ©U2]

where I is the unit matrix of order (2 X 2) and 0

1

0

0

1

0

0

1

0

0

0

0

0

1

0

U1=U2=ZE,s0OErs=

0

Since

ax ax

1

0

0

1

0

0

0

0

0

0

0

0

1

0

0

1

only a simple calculation is necessary to obtain the result. It is found that

a(AOX) ax

a12 0

0

all

0

a12

0

0

0

0

0

0

a22

0

0

a21

0

a22

0

0

0

0

0

0

0

0

0

0

0

0

0

0

a12

0

0

all

0

a12

0

0

0

0

0

0

0

0

a21

0

a22

0

0

a21

0

a22

all

0

0

0

a21

0

0

0

0 all

(il) We evaluate ICS

Y = AOX =

allxll

alixl2

a12x11

a12x12

a11x21

a11x22

a12X21

a12x22

a21 x11

a21 x 12

a22 x 11

a22 x 12

a21x21

a21x22

a22x21

a22x22

and then make use of (6.1) to obtain the above result.

[Ch. 6

6.4 THE CHAIN RULE FOR THE DERIVATIVE OF A MATRIX WITH RESPECT TO A MATRIX We wish to obtain an expression for (0l'0

az

ax

where the matrix Z Is a matrix function of a matrix X, that is

Z = Y(X) where

X = [xii] is of order (m X n) Y = [ yil] is of order (u X v) Z = [zri] is of order (p X q) By definition in (6.1)

az

az

r=1,2,...,m

ax

axrs

s = 1, 2, ... , n

r, s

where Er,s is an elementary matrix of order (m X n),

= r,s

Ers D

i,i

l=1, 2,...,u

azii iiaxrs -

1 = 1, 2, ... , q

where Eli is of order (p X q) As in section 4, 3, we use the chain rule to write az,i

azii

airs

a,

a=1,2,...,u 0=1,2,...,v

ayap

ayap axrs

Hence az

ax =

ayap

Ers

ayap axrs

r, s

ayap axrs

ayap ® az 0e, 9

ax

ayap

O

Ei

azii

(by 2.5)

aya p

(by (4.7) and (4.19))

("1

The Derivative of a Matrix with Respect to a Matrix

88

Sec. 6.4]

The Chain Rule for the Derivative of a Matrix

89

If I,, and It, are unit matrices of orders (n X n) and (p X p) respectively, we can write the above as az

ax

ap

(1-Yli")'& \ IPaYap )

Hence, by (2.11)

M

p (aaX

aX

3z

N) (I.

l\

Yap

Equation (6.5) can be written in a more convenient form, avoiding the summation, if we define an appropriate notation, a generalisation of the previous one. Since

Y1i

Y12

Y21

Y22

LYu1

Yu2

Y =

than (vec Y)' _ y y21

...

Yiv Y2v

...

YuvJ

. Yuv J

We will write the partitioned matrix Laax®1

aXi(3)

P

1;...ax P

P

as

a

ax

or as

a (vec Y)'

ax

®IP

®IP

Similarly, we write the partitioned matrix az

In ® aYii

aY21

az In

ayuv

as

P In®

az l `DIN

az

In Ox -

a vec Y

[Ch. 6

The Derivative of a Matrix with Respect to a Matrix

90

We can write the sum (6.5) in the following order Y11

ax = raax

ray" 01] (1" © P ax yu1 +l

aaZ 1 +

® IPJ CI"

'r4

(0I(0

az

IL

Yzi

1.

..n

+auv®IPI"° azLayx

J[

aZ

ayu.J

We can write this as a (partitioned) matrix product +,G

_)I :,.

az

ayii©I aY21 ax r 75X P* ax 1

P

ax

-

az

I" ®ayuv Finally, using the notations defined above, we have a [vec Y]'

aZ

ax

,,p

az

aZ 1"0 ®

P

L"

(6.6)

a vec Y] fro

We consider a simple example to illustrate the application of the above formula. The example can also be solved by evaluating the matrix Z in terms of the components of the matrix X and then applying the definition in (6.1). w-.

Example 6.6 Given the matrix A = [au] and X = [x11] both of order (2 X 2), evaluate

aziax where Z = Y'Y and Y = AX. (i) Using (6.6) (ii) Using a direct method. Solution (1) For convenience write (6,6) as

az ax = Q

QR

[a[vecYr ®I ax P]

az

N

where

and

R = IO a vec Y

The Chain Rule for the Derivative of a Matrix

Sec. 6.4]

91

From Example 4.8 we know that

ay" ± A'Er ax

so that Q can now be easily evaluated,

Q

I

0

00

a22 0 0

1

a22 0 0

Also in Example 4.8 we have found aZ

= E,S Y + Y'Ers

aYra

we can now evaluate R 2Y11

Y12

0

0

Y12

0

0

0

0

0

2Y11

Y12

o

0

Y12

0

0

0 0

2Y21 Yn

0

0 0 all 0 all 0 0 1 0 0 0 all 001

a21 0 I

Y22

0

0

0

0

2Y21

Nom'

Y22

0

0

Y22

0

0

Y11

0

0

Yil

2Y,2

0

0

0

0

0

Y11

0

0

Y 2Y,2

R =

0""Y21"0""0" Y21

2Y22 0

0

0

0

0

Y21

Lo

0

Y21

2y2

X

000

ate

0 0 a21

0

000

a2i

0 0 a22 0

0 0 a12 0

0

00 0 all 0 0 a12 0 0 0 0 all 0 0 all 0

1

000

a22

I

The Derivative of a Matrix with Respect to a Matrix

92

(Ch.

The product of Q and R is the derivative we have been asked to evaluate

QR =

a11y12 + a21y22

o

0

a11y1 l +a21y21

a12y12 +1122Y22

o

;,c

E2ailyil + 2a21y21 a11y12 + a21y22

2412y + 2a22Y21 La12y12 +a22y22

0

ally,, + a21y21 2a11y12 + 2a21y22

al2y11 + a22y21

a12.y11 + a22y21 2a12y12 + 2a22y22

(ii) By a simple extension of the result of Example 4.6(b) we find that when

Z = X'A'AX az

axrs

= ErSAAX + X'A'AErs

= ErsA'Y + Y'AErs where Y = AX.

By (6.1) and (2.11)

ax

r-.

az

(Ers Ox Ers) (10 A'Y) + 2 (I OO Y'Z) (Ers Ox Ers) r.s

r,s

Since the matrices involved are all of order (2 X 2) 0

0

0

0

0

1

0

0

^'.'

IErsOE;s =

1

1

0

0

0

0

0

1

1

0

0

1

0

0

0

0

0

0

0

1

0

0

and

O--

E Ers OX Ers =

0 1

0

On substitution and multiplying out in the above expression for aZfaX, we obtain the same matrix as in (i). Problems for Chapter 6

(1) Evaluate aYjaX given

y_

[cos (X12 + x22) xux211 X12x22

and

X=

x11

x12

IX-21

X22

Problems .L]

6] (2)

rxil

The elements of the matrix X =

x12 LX13

93

x21

x22 X23 J

are all independent. Use a direct method to evaluate aX/aX.

()3

I x11

x12

x21

x22

Given a non-singular matrix X = _

]

.mar

use a direct method to obtain

ax-1

ax and verify the solution to Example 6.4.

(4) The matrices A = [aiij and X = [x,ij are both of order (2 X 2), X is nonsingular. Use a direct method to evaluate

a(A 0 X-')

ax

CHAPTER 7

Some Applications of Matrix Calculus 7.1 INTRODUCTION

As in Chapter 3, where a number of applications of the Kronecker product were

considered, in this chapter a number of applications of matrix calculus are discussed. The applications have been selected from a number considered in the published literature, as indicated in the Bibliography at the end of this book. These problems were originally intended for the expert, but by expansion and simplification it is hoped that they will now be appreciated by the general reader.

7.2 THE PROBLEMS OF LEAST SQUARES AND CONSTRAINED OPTIMISATION IN SCALAR VARIABLES

In this section we consider, very briefly, the Method of Least Squares to obtain a curve or a line of `best fit', and the Method of Lagrange Multipliers to obtain an extremum of a function subject to constraints. For the least squares method we consider a set of data

i = 1, 2, ..., n

(xi, Yi)

(7.1)

'L7

and a relationship, usually a polynomial function (7.2)

Y = f(x) For each x;, we evaluate f(xi) and the residual or the deviation

ei = y, -f(xr) .

(7.3)

E--

The method depends on choosing the unknown parameters, the polynomial coefficients when f(x) is a polynomial, so that the sum of the squares of the residuals is a minimum, that is n

S = > ei is a minimum.

(Yi -f(x,))'

(7.4)

The Problems of Least Square and Constrained Optimisation

[Sec. 7.21

95

In particular, when f(x) Is a linear function

y =ao+alx S(ao, al) is a minimum when

as as0

C/!

as

(7.5)

=0=as . 1

These two equations, known as normal equations, determine the two unknown parameters ao and a1 which specify the line of 'best fit' according to the principle of least squares. For the second method we wish-to determine the extremum of a continuously differentiable function

f(x1,x2, ...,xn)

(7.6)

whose n variables are contrained by in equations of the form

g1(x1,x2,...,x,) = 0,

1 = 1,2,...,rr

The method of Lagrange Multipliers depends on defining an augmented function

ff+

m

1pigt t=1

where the pi are known as Lagrange multipliers.

The extreme of f(x) is determined by solving the system of the (m + n) equations

af* ax,

=a

g; = 0

r = 1, 2, .. , n

i = 1,2,...,m

for the m parameters µl, u2, ... , µm and the n variables x determining the extremum. Example 71

Given a matrix A = [a11] of order (2 X 2) determine a symmetric matrix X = [x;j] which is a best approximation to A by the criterion of least squares. Solution Corresponding to (7.3) we have

E=A - X where E = [e;1] and e11 = a;i -x1j.

96

Some Applications of Matrix Calculus

[Ch. 7

.ti

The criterion of least squares for this example is to minimise

S=e= l,/

which is the equivalent of (7.6) above. The constraint equation is

Xi2 -x21 = 0 and the augmented function is

f* = Earl -x1/)2 + µ(x12 -x21) = 0

-2(a ll '-x11)

ax11

af*

-2(a12 -x12) +',1 = 0

ax12

af*

- -2 (a21 -x21) -11 = 0

.N+

ax21

= 0

af*

-2 (a22 - x22) = 0

ax22

This system of 5 equations (including the constraint) leads to the solution

µ = a12 -x21

x11 = all , x22 = a22 , x12 = x21 = J(a12 + a21) Hence a12 + a21

all

2

X =

2

a12 + a21

L

all

a12

a21

a22

+ 2

all

a21

a12

a22

a22

2

= j(A+A') 7.3 PROBLEM 1 - MATRIX CALCULUS APPROACH TO THE PROBLEMS OF LEAST SQUARES AND CONSTRAINED OPTIMISATION

If we can express the residuals in the form of a matrix E, as in Example 7.1, then the sum of the residuals squared is

S = tr E'E

.

(7.10)

Problem 1

Sec. 7.3]

97

The criterion of the least squares method is to minimise (7,10) with respect to the parameters involved.

The constrained optimisation problem then takes the form of finding the matrix X such that the scalar matrix function

S = f(X) is minimised subject to contraints on X in the form of

.G(X)=0

(7.11)

where G = [gill is a matrix of order (s X t) where s and t are dependent on the a.-

number of constraints g1l involved.

As for the scalar case, we use Lagrange multipliers to form an augmented matrix function f*(X). Each constraint gil is associated with a parameter (Lagrange multiplier) Ail Since

where

Eµllg;l = tr U'G

U = [µtl]

we can write the augmented scalar matrix function as

f*(X) = trE'E+ tr U'G

(7.12)

which is the equivalent to (7.8). To find the optimal X, we must solve the system of equations

af* = 0. ax

(7.13)

Problem

Given a non-singular matrix A = [ail] of order (n X n) determine a matrix X = [x,1] which is a least squares approximation to A

(i) when X is a symmetric matrix (ii) when X is an orthogonal matrix. Solution (i) The problem was solved in Example 7.1 when A and X are of order (2 X 2). With the terminology defined above, we write

E=A - X G(X) = X -X' = 0 so that G and hence U are both of order (n X n).

Some Applications of Matrix Calculus

98

[Ch. 7

Equation (7.12) becomes

f* = trA'A-trA'X-trX'A+trX'X+trU'X-trU'X'. We now make use of the results, in modified form if necessary, of Examples 5.4 and 5.5, we obtain

of ax

_ -2A+2X+U-U' = 0

for X = A+

U °- U' 2

Then

X'=A'+U'-U 2

and since X = X', we finally obtain `""

X=j(A+A'). E'"

(ii) This time

G(X)=X'X-I=0. Hence

f* = tr[A'-X'][A-X] +trU'[XX'-I]

so that

af

ax

_ -2A+2X+X[U+U']

=0 for X=A-X

2

fl.

Premultiplying by X' and using the condition

X'X = I we obtain =I+U+U'

X'A

2

and on transposing

A'X = I+

U+ U' 2

Hence

A'X = X'A

.

(7.14)

,_, ...

If a solution to (7.14) exists, there are various ways of solving this matrix equation.

Sec. 7.3]

Problem 1

99

For example with the help of (2.13) and Example (2.7) we can write it as

[(l ©A') .- (A' ©I)U] x = 0

(7,15)

where U is a permutation matrix (see (2.24)) and

x=vecX. .D.

We have now reduced the matrix equation into a system of homogeneous ...

equations which can be solved by a standard method. If a non-trivial solution to (7.15) does exist, it is not unique. We must scale it appropriately for X to be orthogonal.

There may, of course, be more than one linearly independent solution to (7.15). We must choose the solution corresponding to X being an orthogonal matrix.

Example 72 Given

A =

find the othogonal matrix X which is the least squares best approximation to A. Solution

-1

0

2

1

0

0

0

0

1

-1

0

0

2

1

[IOA'] =

r1 -1

0

and [A'©1]U =

1

0

0

0

0

1 -1

2

1

0

0

0

0

2

1

Equation (7.15) can now be written as 0

0

0

0

2

1

-1

1

-2 -1

1

-1

0

0

0 1-+

0

x=0

'L7

There are 3 non-trivial (linearly independent) solutions, (see [18] p.131). They are

x = [1 -2 1 1]',

x = [1

1

2 -1]'

and

Only the last solution leads to an orthogonal matrix X, it is

X=1

13

2

3

-3

2

x = [2 -3 3 2]'.

[Ch. 7

Some Applications of Matrix Calculus

100

7.4 PROBLEM 2 - THE GENERAL LEAST SQUARES PROBLEM The linear regression problem presents itself in the following form: N samples from a population are considered. The ith sample consists of an te/

observation from a variable Y and observations from variables X1, X2, ..., X (say).

We assume a linear relationship between the variables. If the variables are measured from zero, the relationship is of the form

Yl = bo+blxn+b2x11+...+bx,8+el.

(7.16)

If the observations are measured from their means over the N samples, then

(i= 1, 2, ... N)

yr =

(7.17)

bo, b1, b2, ... , b are estimated parameters and e1 Is the corresponding residual. In matrix notation we can write the above equations as

y = Xb + e

(7.18)

[]

where

Y=

.

b=

ba

,

eI

e=

2

...

Y2

[bl]

YNI' and

rl

... xln

X22 ... X2n

X11 X12 or

X =

... Xln

X21 X22 ... x2n

...

1

...

I{

x12

ex

...

X =_

Ibn

L1

XN2 ... XNnJ

LXNI XN2 ... XNnJ

.

As already indicated, the `goodness of fit' criterion is the minimisation with respect to the parameters b of the sum of the squares of the residuals, which in this case is

S = e'e = (y'-b'X')(y-Xb).

Making use of the results in table (4.4), we obtain a (e'e)

ab

=

-(y

'X)'-X'y + (X'Xb +X'Xb)

= -2X'y + 2X'Xb = 0 for X'Xb = X'y.

(7.19)

where b is the least squares estimate of b. If (X'X) is non-singular, we obtain from (7.19) b

= (X'X)-1 X'y..

(7.20)

Problem 2

Sec. 7.41

101

We can w,ite (7.19) as

X'(y -Xi) = 0 X'e = 0

or

(7.21)

which is the matrix form of the normal equations defiend in section 7.2. Example Z 3

Obtain the normal equations for a least squares approximation when each sample consists of one observation from Y and one observation from

(i) a random variable X (ii) two random variables X and Z. Solution (1)

X =

Y,

x1

1

x2

1

I

y =

Y2

6, ,

b = 62

... 1

XN

YN

hence

X'[y-Xb] = Ey;-b1N-b2Ex; ExiYi - b, Ex; - 62 Ex,2J So that the normal equations are .-0

Ey, = b,N+b2Ex1 and Exly! = b1 E xr + b2 Ex,? .

(ii) In this case x1 l

x2 z2

bl

Y11

y =

b=

Y2

...

...

X=

z

Lb3J

11 xN ZNJ

LYNJ

The normal equations are