9789144100968

Victor Ufnarovski was educated at Moscow University. He is a professor of Mathematics at Lund University. His main research is Algebra, in particular matrix theory and computer algebra. For many years he has been training Swedish high school students for the International Mathematical Olympiad.

Matrix Theory

This book is based on the course Matrix theory given at Lund University.

It starts by recalling the basic theory of matrices and determinants, and then proceeds to more advanced subjects such as the Jordan Normal Form, functions of matrices, norms, normal matrices and singular values.

The book may be used for a second course in linear algebra in a bachelor’s program in mathematics, or for a first year graduate course in engineering subjects.

Matrix Theory

| Matrix Theory

A good understanding of matrices and their properties is a necessary prerequisite for progress in almost any field within pure or applied mathematics, for example calculus in several variables, numerical analysis or control theory.

Anders Holst Victor Ufnarovski

Anders Holst is a lecturer in Mathematics at Lund University, Sweden. His research interests belong to the fields of Fourier Analysis and Partial Differential Equations. Currently he is the Director of Studies at the department for Mathematics at the Faculty of Engineering.

Distinguishing qualities compared with other texts on the subject are • The book starts gently compared with other texts with the same scope.

• There are many carefully worked out examples.

• It is possible to use the book for classes on different levels, by selecting parts of the material.

The book contains a large number of exercises with answers, indication of difficulty and sometimes hints. The exercises are intended to help the students to think in new ways and to understand the art of proving mathematical statements.

Art.nr 38585 ISBN 978-91-44-10096-8

www.studentlitteratur.se

978-91-44-10096-8_01_cover.indd 1

Anders Holst Victor Ufnarovski

9 789144 100968

2014-08-13 15:24

Copying prohibited All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

The papers and inks used in this product are eco-friendly.

Art. No 38585 ISBN 978-91-44-10096-8 Edition 1:1 Š The Authors and Studentlitteratur 2014 www.studentlitteratur.se Studentlitteratur AB, Lund

Cover design by: Francisco Ortega Printed by Eurographic Danmark A/S, Denmark 2014

INTRODUCTION

This course is a thorough introduction to matrix theory. The reader has probably already taken a course in linear algebra, and is familiar with matrices and their basic properties. But here we plan to go much further.

During the first few weeks of the course, it is common for students to get the impression that they have suddenly lost their understanding of matrices: things that seemed familiar suddenly look strange and very complicated.

The reason is that different problems demand different points of view, and matrices may be presented in quite different ways in order to fit in with these different perspectives.

This variety of points of view and ways of presenting matrices is one of the most difficult parts of the course, but at the same time one of the most valuable lessons for future applications because this variety gives the student many different useful approaches to the same object – the matrix.

The book is written on the basis of the course in matrix theory at LTH – the Faculty of Engineering at Lund University. We are grateful to our colleagues Per Alexandersson, Jörgen Backelin, Olof Bergvall, Lars-Christer Böiers, Andrey Ghulchak, Jan Gustavsson, Johan Helsing, Arne Meurman, Pelle Pettersson, Sergei Silvestrov, Gunnar Sparr, Martin Tamm, Lars Vretare and Yishao Zhou with whom we discussed the course and this book. We are especially thankful to Sven Spanne who wrote the first book for this course and formed its structure and level. The original problems from his exercise book are so interesting (and so difficult) that we without hesitation included most of them here. John Jones helped us to reduce the number of grammatical errors. Last but not least, we want to thank our main readers – the LTH students (especially Waqar Hameed) who helped us to improve this book by their unexpected questions, expected criticism and useful suggestions.

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CONTENTS

CHAPTER 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

Matrices: long and short notations 9 Block matrices 11 Matrices as numbers. Addition and subtraction 15 Multiplication by numbers. Some special classes of matrices 16 Matrix multiplication 18 Multiplication by elementary matrices 22 Associativity. Powers. Nilpotent matrices 24 Non-commutativity. Left and right inverse 27 Exercises 31 Hints and answers 33

CHAPTER 2

2.1 2.2 2.3 2.4 2.5

4

Gaussian Elimination and LU-decomposition 37

Echelon matrices. Pivot elements. Free and basic variables 37 Gaussian elimination. LU-decomposition 39 Rank 44 Exercises 48 Hints and answers 50

CHAPTER 3

3.1 3.2 3.3 3.4

Matrices – Conventions and Notations 9

Determinants 53

Determinant: notation and recursive definition 53 Determinants: triangular,permutation matrices 55 Linear and alternating properties 57 Gaussian elimination in ∣A∣. Abstract definition 61 © T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

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3.5 3.6 3.7 3.8 3.9 3.10

The determinant of a product. Expansion along an arbitrary column 63 Even and odd permutations. Complete expansion. ∣AT ∣ 65 Expansion along a row. The inverse matrix 69 How to calculate the determinant. Vandermonde 72 Exercises 77 Hints and answers 80

CHAPTER 4

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

The definition of field 83 The definition of vector space 84 Linear independence. Generating sets. Dimension 85 Coordinates in different bases. The transition matrix 88 The bases for a space and its subspaces 89 Sum and intersection of subspaces. Direct sum 92 How to find a basis 93 Exercises 100 Hints and answers 102

CHAPTER 5

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Linear Maps 105

Definition and examples of linear maps 105 The matrix for a linear map 106 Matrix in a new basis. Similar matrices 110 Determinant, trace and rank of an operator. Image 113 Kernel 118 Dimension arguments 122 Example: some field theory 127 Exercises 136 Hints and answers 138

CHAPTER 6

6.1 6.2 6.3

Finite Dimensional Vector Spaces 83

Spectral Theory 141

Diagonalization. Eigenvectors and eigenvalues 141 Characteristic polynomial. Diagonalization 148 Diagonalizable matrices 152

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6.4 6.5 6.6

Non-diagonalizable matrices 158 Exercises 160 Hints and answers 162

CHAPTER 7

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Invariant subspaces 165 Nilpotent operators 169 Uniqueness. Constructing the Jordan form 175 Constructing the basis for the normal form 181 The Jordan normal form 193 An example of Jordanization 197 Proving theorems using the Jordan form 207 Exercises 211 Hints and answers 213

CHAPTER 8

8.1 8.2 8.3 8.4 8.5

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Functions on Matrices 233

How to define f (A) 233 Calculating f (A) using polynomials 239 Lagrange interpolation 241 Hermite interpolation 245 Exercises 249 Hints and answers 250

CHAPTER 10

10.1 10.2

The Minimal Polynomial 217

The Cayley-Hamilton theorem 217 The minimal polynomial 220 Jordan decomposition 225 Exercises 227 Hints and answers 229

CHAPTER 9

9.1 9.2 9.3 9.4 9.5 9.6

The Jordan Normal Form 165

Inequalities and Positive Matrices 253

Inequalities and their transformations 253 Some useful inequalities 257 Š T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

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10.3 10.4 10.5 10.6 10.7

Inequalities with complex numbers 263 Positive matrices 266 Graphs and page ranking 270 Exercises 273 Hints and answers 274

CHAPTER 11

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Compact sets and continuous functions 275 Norms 279 Matrix and operator norms 284 The condition number 290 The spectral radius 292 Exercises 295 Hints and answers 296

CHAPTER 12

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10

Inner Products and Orthogonality 299

Hermitian conjugation 299 Inner products and orthogonal bases 301 Getting a norm from the inner product 306 Unitary matrices and QR-factorization 308 The adjoint operator 314 Hermitian matrices 316 Orthogonal projection and outer product 318 Infinite dimensional vector spaces 321 Exercises 325 Hints and answers 326

CHAPTER 13

13.1 13.2 13.3 13.4 13.5

Norms 275

Singular Values 329

Singular values 329 Schur’s lemma 337 Normal matrices 341 Exercises 345 Hints and answers 348

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CHAPTER 14

14.1 14.2 14.3 14.4 14.5 14.6 14.7

Quadratic forms and their matrices 353 Hermitian forms 355 Diagonalization of Hermitian forms 359 Congruent matrices 365 Positive definite matrices 368 Exercises 375 Hints and answers 378

CHAPTER 15

15.1 15.2 15.3 15.4 15.5

Quadratic and Hermitian forms 353

The Moore-Penrose Pseudoinverse 381

Definition of the Moore-Penrose pseudoinverse 381 Calculating the Moore-Penrose pseudoinverse 385 The least squares method 390 Exercises 394 Hints and answers 395

Bibliography 397

Index 399

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CHAPTER 3

Determinants

The determinant of a square matrix A is a number which gives important information about this matrix, first of all whether it is invertible or not. We do not define determinants of non-square matrices so in this chapter all matrices are square matrices. The determinant may be considered as a function of the matrix elements. This function is quite complicated and can be defined in different equivalent ways. Our approach is as follows. First of all we give a recursive definition, which is relatively easy to understand but unsuitable for practical calculations, except for small matrices. Using this definition we will study the main properties of determinants. Those properties will allow us to calculate the determinant more practically and efficiently, and will also lead us to other equivalent definitions. The reader can choose his/her own definition which reflects his/her understanding of the determinants best.

3.1 Determinant: notation and recursive definition First of all we need a convenient notation for the determinant. If A is a square matrix one safe notation is det(A). This notation is convenient if we have a name for the matrix, but clumsy if we need to express the matrix A in terms of its rows or elements. In such cases it is more common to use the notation ∣A∣, which in terms of the matrix elements takes the form RRR RRR RRR RRR RRR RRR RRR RRR RR

a 11 a 12 a 13 ⋯ a 1n a 21 a 22 a 23 ⋯ a 2n a 31 a 32 a 33 ⋯ a 3n ........................ a n1 a n2 a n3 ⋯ a nn

RRR RRR RRR RRR RRR .

RRR RRR RRR RR

This notation has a disadvantage for matrices of size 1: we cannot distin© T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

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guish the determinant ∣a∣ from the absolute value ∣a∣. These numbers are different in general. For example ∣ − 2∣ = −2 as determinant and ∣ − 2∣ = 2 as absolute value. Fortunately, we normally work with larger matrices, in which case the notation must stand for the determinant. In special cases when confusion might occur we will use unambiguous notations such as det(a), abs(a) or (better!) words such as “the matrix a,” or “the number a”, which should resolve possible ambiguities. This is a small price to pay for the very convenient notation ∣A∣.

We construct our definition recursively, starting from size 1 and then using induction to define the determinant of a matrix of size n in terms of determinants of submatrices of the smaller size n − 1.

For matrices of size one the definition is simple: if A = (a) then ∣A∣ = a.

In other words, if the matrix is a number this number is its own determinant.

Suppose now that we already know how to calculate the determinant for matrices of size n − 1 and want to define the determinant ∣A∣ for a matrix of size n > 1. This will be more complicated. First of all we introduce some notations. For any indices i, j consider the submatrix obtained from A by deleting row i and column j. This submatrix has size n−1 and we can calculate its determinant which we denote by D i j . Now we can write our recursive definition: ∣A∣ = a 11 D 11 − a 21 D 21 + a 31 D 31 − ⋯ ± a n1 D n1 , where we alternate signs, so the last one will be + for n odd and − for n even.

The reader hardly feels a great enthusiasm if seeing this definition for the first time, so let us try to understand it better. Remember, we only need some definition, to show the existence of the determinant. We do not pretend to understand it now – we will do that later. We only want to understand how this definition works and if it possible to use it in practice. Let us try the case n = 3 and suppose that we already know how to calculate the determinant for 2 × 2 matrices (which in fact we know from linear algebra).

Let ⎛ a 11 a 12 a 13 ⎞ ⎟ A=⎜ ⎜ a 21 a 22 a 23 ⎟ . ⎝ a 31 a 32 a 33 ⎠ First we need to calculate the three numbers D 11 ,D 21 ,D 31 . As to D 11 we need to delete the first row and the first column of A and calculate the deter54

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minant of the obtained matrix: D 11 = ∣

a 22 a 32

a 23 ∣ a 33

For D 21 we need to delete the second row and the first column (again) and calculate the determinant: D 21 = ∣

a 12 a 32

a 13 ∣. a 33

Working in the same way we get the third number D 31 . Finally, we obtain the formula for the determinant ∣A∣: ∣A∣ = a 11 D 11 − a 21 D 21 + a 31 D 31 , ∣A∣ = a 11 ∣

a 22 a 32

a 23 a 12 ∣ − a 21 ∣ a 33 a 32

a 13 a 12 ∣ + a 31 ∣ a 33 a 22

a 13 ∣. a 23

In this form we see rather clearly that all we need to calculate the determinant of a 3 × 3 matrix A is to be able to calculate arbitrary 2 × 2 determinants.

What about these? Using the same approach we find that ∣

a c

b ∣ = a∣d∣ − c∣b∣ = ad − cb = ad − bc, d

because we know that for the matrices d and b, of size 1, their determinants ∣d∣,∣b∣ are equal to their values d,b (and do not mix up the determinant ∣d∣ = det(d) with the absolute value ∣d∣ = abs(d)).

Of course, after this we can even get the complete expansion for the 3 × 3 determinant, something like ∣A∣ = a 11 (a 22 a 33 − a 23 a 32 ) − a 21 (a 12 a 33 − a 13 a 32 ) + a 31 (a 12 a 23 − a 13 a 22 ) = a 11 a 22 a 33 − a 11 a 23 a 32 − a 12 a 21 a 33 + a 13 a 21 a 32 + a 12 a 23 a 31 − a 13 a 22 a 31 , but we have no interest in it just now. The important thing is that in principle we have a way to calculate the determinant of any square matrix of arbitrary size n.

3.2 Determinants of triangular and permutation matrices How practical is the method of calculating the determinant using this recursive definition? We see that for size n we may need n times more computations © T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

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than for size n − 1. So the complexity grows at least as n! and this is hardly practical for general matrices. But if the matrix has many zeros this can be quite practical.

Theorem 3.1 If A is a triangular matrix then its determinant is equal to the product of the numbers on the main diagonal ∣A∣ = a 11 a 22 ⋯a nn .

Proof. This is a good exercise to understand how the induction works. We will consider upper triangular matrices only, commenting on the proof for the lower triangular matrices much later; see the end of section 3.6.

The induction starts easily: for matrices of size 1 there is nothing to prove.

If the size is an arbitrary n > 1 then using the recursive definition we see that ∣A∣ = a 11 D 11 − 0D 21 + 0D 31 − ⋯ ± 0D n1 = a 11 D 11 .

It only remains to note that the matrix obtained by deleting the first row and the first column from A is upper triangular as well, so by the induction hypothesis its determinant is equal to the product of the numbers on the main diagonal, in other words D 11 = a 22 a 33 ⋯a nn and together we get ∣A∣ = a 11 ⋅ a 22 a 33 ⋯a nn , ◻

as desired.

For example, if i ≠ j then ∣I + cE i j ∣ = 1 for any elementary matrix.

Theorem 3.2 If A is a permutation matrix then its determinant is equal to ±1.

Proof. Again induction works perfectly. The case n = 1 is trivial, as usual. For the induction step n − 1 ⇒ n note that in the recursive definition only one term in the sum is non-zero, because only one element in the first column is different from zero, say a k1 = 1. Then ∣A∣ = ±a k1 D k1 = ±D k1 and it only remains to note that the matrix corresponding to the determinant D k1 also is a permutation matrix. ◻ 56

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3.3 Linear and alternating properties of the determinant To understand the determinant better we shall now study how it behaves under simple row operations.

Theorem 3.3 If we multiply some row with a constant c then the determinant is also multiplied by the same constant: RRR RRR RRR RRR RRR RRR RRR RRR RR

A1 ⋯ cA i ⋯ An

RRR RRR RRR RRR RRR RRR RRR R RRR = c RRRRR RRR RRR RRR RRR RRR RRR RR RR

A1 ⋯ Ai ⋯ An

RRR RRR RRR RRR RRR .

RRR RRR RRR RR

Proof. As usual we use induction, the case n = 1 being trivial. We consider the case of the first row only, because the other rows can be considered similarly.

RRR RRR RRR RRR RRR RRR RRR RRR RR

ca 11 ca 12 ca 13 ⋯ ca 1n a 21 a 22 a 23 ⋯ a 2n a 31 a 32 a 33 ⋯ a 3n ........................... a n1 a n2 a n3 ⋯ a nn RRR RRR RR −a 21 RRRR RRRR RRR R

RRR RRR RRR RRR RRR RRR R RRR = ca 11 RRRRR RRR RRR RRRR RRRR RRR R R

RRR RRR RRR RRR RRR R RRR + a 31 RRRRR RRR RRRR RRR RRR RR R

ca 12 ⋯ ca 1n a 32 ⋯ a 3n .............. a n2 ⋯ a nn

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

ca 12 ⋯ ca 1n a 22 ⋯ a 2n .............. a n2 ⋯ a nn

RRR RRR RRR RRR − RRR RRR RR

RRR RRR RRR RRR − . . . = RRRR RRR R

(using the induction hypothesis) RRR RRR RR ca 11 RRRR RRRR RRR R

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

R ⎛ RRRR ⎜ RRRR c⎜ ⎜a 11 RRRR ⎜ RR ⎝ RRRR R

RRR RRR RRR RRR RRR R RRR−ca 21 RRRRR RRR RRRR RRR RRR R RR

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

a 12 ⋯ a 1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

RRR RRR RRR RRR RRR R RRR − a 21 RRRRR RRR RRRR RRR RRR RR R

RRR RRR RRR RRR RRR R RRR+ca 31 RRRRR RRR RRRR RRR RRR R RR

a 12 ⋯ a 1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

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a 12 ⋯ a 1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

RRR RRR RRR RRR RRR R RRR + a 31 RRRRR RRR RRRR RRR RRR RR R

RRR RRR RRR RRR−. . . = RRRR RRR R

a 12 ⋯ a 1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

RRR RRR ⎞ RRR ⎟ RRR . . .⎟ ⎟= RRRR ⎟ RRR ⎠ R 57

3 determinants

RRR RRR RRR RR = c RRRR RRR RRR RRR RR

RRR RRR RRR RRR RRR .

RRR RRR RRR RR

◻

RRR RRR RRR RRR RRR = 0.

RRR RRR RRR RR

◻

a 11 a 12 a 13 ⋯ a 1n a 21 a 22 a 23 ⋯ a 2n a 31 a 32 a 33 ⋯ a 3n ........................ a n1 a n2 a n3 ⋯ a nn

An evident but important corollary is Corollary 3.4 If A has a zero row then ∣A∣ = 0.

Proof.

RRR RRR RRR RRR RRR RRR RRR RRR RR

A1 ⋯ O ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR = RRR RRR RRR RRR RRR RRR RRR RR RR

A1 ⋯ 0O ⋯ An

RRR RRR RRR RRR RRR RRR RRR R RRR = 0 RRRRR RRR RRR RRR RRR RRR RRR RR RR

A1 ⋯ O ⋯ An

Another important property with a similar proof is Theorem 3.5 If some row can be written as a sum A i = A′i + A′′i then the determinant is the sum of the corresponding determinants: RRR A1 RRR RRR ⋯ RRR RRR A′i + A′′i RRR ⋯ RRR RRR A n RR

RRR RRR RRR RRR RRR RRR RRR RRR RRR = RRR RRR RRR RRRR RRRR RRR RRR R R

A1 ⋯ A′i ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRRR RRRR RRR RRR R R

A1 ⋯ A′′i ⋯ An

RRR RRR RRR RRR RRR .

RRR RRR RRR RR

Proof. Again, we use induction and only consider the case of the first row.

RRR a′ + a ′′ a ′ + a ′′ a′ + a ′′ ⋯ a ′ + a′′ 11 12 12 13 13 1n 1n RRR 11 RRR a a a ⋯ a 21 22 23 2n RRR RRR a 31 a 32 a 33 ⋯ a 3n RRR RRR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

RRR a n2 a n3 ⋯ a nn RR a n1 58

RRR RRR RRR RRR RRR = RRR RRR RRR RR

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RRR RRR R ′ ′′ RRR (a 11 + a 11 ) RR RRR RRR RR

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

RRR RRR a ′ + a ′′ ⋯ a ′ + a ′′ 12 1n 1n RRR RRR 12 RRR RRR a 32 ⋯ a 3n RRR − a 21 RRR RRR RRR . . . . . . . . . . . . . . . . . . . . . . .

RRR RRR a n2 ⋯ a nn RR RR

RRR a ′ + a′′ ⋯ a ′ + a ′′ 12 1n 1n RRR 12 RRR a 22 ⋯ a 2n +a 31 RRR RRR . . . . . . . . . . . . . . . . . . . . . . .

RRR a n2 ⋯ a nn RR

RRR RRR RRR RRR + RRR RRR RR

RRR RRR RRR RRR − . . . = RRR RRR RR

(using the induction hypothesis) RRR RRR R ′ ′′ RRR (a 11 +a 11 ) RR RRR RRR RR

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn R ⎛RRRR ⎜RRRR +a 31 ⎜ ⎜RRRR ⎜RR ⎝RRRR R

RRR RRR R ′ RRR a 11 RR RRRR RRR R

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

RRR RRR R ′′ RRR +a 11 RR RRR RRR RR

RRR RRR RRR RRR RRR R RRR − a 21 RRRRR RRR RRRR RRR RRR R RR RRR RRR RRR RRR RRR R RRR −a 21 RRRRR RRR RRR RRR RRR RR RR

a ′12 ⋯ a ′1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RR RR

a ′12 ⋯ a′1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

a 22 ⋯ a 2n a 32 ⋯ a 3n ............. a n2 ⋯ a nn RRR RRR RRR RRR RRR RRR RRR RRR RR

RRR R ⎛RRRR RRR RRR ⎜RRRR RRR−a 21 ⎜ ⎜RRRR RRR ⎜RR RRR ⎝RRRR RR R

a ′′12 ⋯ a ′′1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

a ′12 ⋯ a ′1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn a ′′12 ⋯ a ′′1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

a ′11 a ′12 ⋯ a ′1n a 21 a 22 ⋯ a 2n a 31 a 32 ⋯ a 3n ................... a n1 a n2 ⋯ a nn

RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRRR RRRR RRR RRR R R

RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RR RR

a ′′12 ⋯ a ′′1n a 32 ⋯ a 3n ............. a n2 ⋯ a nn

RRR RRR⎞ RRR⎟ RRR⎟ + RRR⎟ ⎟ RRR⎠ RR

RRR RRR⎞ RRR⎟ RRR⎟ −⋯= RRR⎟ ⎟ RRR⎠ RR

RRR RRR RRR RRR RRR R RRR + a 31 RRRRR RRR RRRR RRR RRR R RR

a ′12 ⋯ a′1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

RRR RRR RRR RRR − . . . + RRRR RRR R

RRR RRR RRR RRR RRR R RRR +a 31 RRRRR RRR RRR RRR RRR RR RR

a ′′12 ⋯ a′′1n a 22 ⋯ a 2n ............. a n2 ⋯ a nn

RRR RRR RRR RRR −. . . = RRR RRR RR

a′′11 a ′′12 ⋯ a ′′1n a 21 a 22 ⋯ a 2n a 31 a 32 ⋯ a 3n ................... a n1 a n2 ⋯ a nn

RRR RRR RRR RRR RRR .

RRR RRR RRR RR

◻

What we have proved in these theorems is that the determinant as a function of its i:th row is a linear function. Because this is true for every row, one can © T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

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express this in a shorter way – the determinant is a multilinear function of its rows. Note that the determinant ∣A∣ is not a linear function of the matrix A for size > 1: ∣A + B∣ ≠ ∣A∣ + ∣B∣, ∣cA∣ ≠ c∣A∣.

But, using the linear property for every row, we can at least say that ∣cA∣ = c n ∣A∣.

The determinant has another important property – it is a skew-symmetric function (one can also say alternating function) of its rows, which means the following.

Theorem 3.6 If two rows in A are equal then ∣A∣ = 0. If we exchange two rows, the determinant changes its sign: RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Ai ⋯ Aj ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR R RRR = − RRRRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Aj ⋯ Ai ⋯ An

RRR RRR RRR RRR RRR RRR RRR .

RRR RRR RRR RRR RRR

Proof. Again induction, but now it is starts from n = 2, where it is simple to check: ∣

a a

b a ∣ = ab − ba = 0; ∣ b c

b c ∣ = ad − bc = −(cb − da) = − ∣ d a

d ∣. b

For the induction step note first that the second statement follows from the first. Indeed, if we know that the determinant with two equal rows is zero then we have, using the linear property twice that RRR RRR RRR A1 A1 RRR RRR RRR RRR R R ⋯ ⋯ RRR RRR RRR RRR A i + A j RRRRR RRRRR Ai R RRR RRR 0 = RRRR ⋯ ⋯ RRR = RRR RRR RRR A i + A j RRRRR RRRRR A i + A j RRR RRR RRR ⋯ ⋯ RRR RRR RRR RRR A RRR RRR A n n R R R 60

RRR RRR A1 RRR RRR RRR RRR ⋯ RRR RRR RRR RRR Aj RRR RRR ⋯ RRR + RRR RRR RRR A + A j RRR RRR i RRR RRR ⋯ RRRR RRRR RR RR A n

RRR RRR RRR RRR RRR RRR RRR = RRR RRR RRR RRR RRR

© T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

3 determinants

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Ai ⋯ Ai ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Ai ⋯ Aj ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Aj ⋯ Ai ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Aj ⋯ Aj ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR = RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Ai ⋯ Aj ⋯ An

RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR + RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR RRR

A1 ⋯ Aj ⋯ Ai ⋯ An

RRR RRR RRR RRR RRR RRR RRR , RRR RRR RRR RRR RRR

which is equivalent with the second statement. Thus it is sufficient to prove the first statement and it is sufficient to prove it for the case j = i + 1 because then we can extend it using the second one. (Try to understand how to do this correctly without logical errors. One way is to use induction on j − i: if we know how the determinant changes when we exchange two consecutive rows, we can then use induction to understand the result of the exchange of arbitrary rows. For example, the exchange of rows 1 and 3 is may be obtained by the exchange of 1 and 2 followed by the exchange of 2 and 3 and after that the exchange of 1 and 2 again.)

We restrict ourselves to the case i = 1, j = 2, because the proof is similar for other pairs i,i + 1. So, suppose that the first and the second rows are equal.

In the recursive definition ∣A∣ = a 11 D 11 − a 21 D 21 + a 31 D 31 − . . . all determinants D 31 , D 41 , . . . are equal to zero by the induction hypothesis, because their matrices contain two equal rows (the first and the second ones).

It remains to note that a 11 = a 21 and that D 11 = D 21 because they are the determinants of equal matrices. ◻

3.4 Gaussian elimination in the determinant. Abstract definition of the determinant The following theorem is the key to efficient calculation of determinants.

Theorem 3.7 If i ≠ j then adding a multiple of row j to row i does not change the determinant. In matrix form: ∣(I + cE i j )A∣ = ∣A∣.

© T H E A U T H O R S A N D S T U D E N T L I T T E R AT U R

61

Victor Ufnarovski was educated at Moscow University. He is a professor of Mathematics at Lund University. His main research is Algebra, in particular matrix theory and computer algebra. For many years he has been training Swedish high school students for the International Mathematical Olympiad.

Matrix Theory

This book is based on the course Matrix theory given at Lund University.

It starts by recalling the basic theory of matrices and determinants, and then proceeds to more advanced subjects such as the Jordan Normal Form, functions of matrices, norms, normal matrices and singular values.

The book may be used for a second course in linear algebra in a bachelor’s program in mathematics, or for a first year graduate course in engineering subjects.

Matrix Theory

| Matrix Theory

A good understanding of matrices and their properties is a necessary prerequisite for progress in almost any field within pure or applied mathematics, for example calculus in several variables, numerical analysis or control theory.

Anders Holst Victor Ufnarovski

Anders Holst is a lecturer in Mathematics at Lund University, Sweden. His research interests belong to the fields of Fourier Analysis and Partial Differential Equations. Currently he is the Director of Studies at the department for Mathematics at the Faculty of Engineering.

Distinguishing qualities compared with other texts on the subject are • The book starts gently compared with other texts with the same scope.

• There are many carefully worked out examples.

• It is possible to use the book for classes on different levels, by selecting parts of the material.

The book contains a large number of exercises with answers, indication of difficulty and sometimes hints. The exercises are intended to help the students to think in new ways and to understand the art of proving mathematical statements.

Art.nr 38585 ISBN 978-91-44-10096-8

www.studentlitteratur.se

978-91-44-10096-8_01_cover.indd 1

Anders Holst Victor Ufnarovski

9 789144 100968

2014-08-13 15:24