Algebra by michael artin

ALGEBRA

Michael Artin
Massachusetts Institute of Technology

UP

DILlMAN

COLLEGE OF SCIENCE
CENTRAL LIBRARY

1111111111111111111111111111111111111111111111111111111111111111111111

UDSCB0035140

II
=-

PRENTICE HALL
Upper Saddle River, New Jersey 07458

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Library of Congress Cataloging-in-Publication Data
Artin, Michael.
Algebra I Michael Artin.
p. em.
Includes bibliographical references and index.

ISBN 0-13-004763-5
1. Algebra.
I. Title.
QAI54.2.A77 1991
512.9-dc20

91-2107
CIP

Figure 4.16 from Zeitschrift fur Kristallographie
EditoriallProduction Supervision and
Interior Design: Ruth Cottrell
Prepress Buyer: Paula Massenaro
Manufacturing Buyer: Lori Bulwin

(:?/9?

UzC;

II
© 1991 by Prentice-Hall, Inc.
A Simon & Schuster Company
Upper Saddle River, New Jersey 07458

All rights reserved. No part of this book may be
reproduced, in any form or by any means,
without permission in writing from the publisher.

90000>
Printed in the United States of America

10 9 8

ISBN 0-13-004763-5
Prentice-Hall International (UK) Limited, London
Prentice-Hall of Australia Pty. Limited, Sydney
Prentice-Hall Canada Inc., Toronto
Prentice-Hall Hispanoarnericana, S.A., Mexico
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Simon & Schuster Asia Pte. Ltd., Singapore
Editora Prentice-Hall do Brasil. Ltda., Rio de Janeiro

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9 780130 047632


To my wife Jean

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Contents

xiii

Preface
A Note for the Teacher
Chapter 1

xv

Matrix Operations

1

1. The Basic Operations 1
2. Row Reduction 9
3. Determinants 18
4. Permutation Matrices 24
S. Cramer's Rule 28
EXERCISES 31

Chapter 2

Groups

38

1. The Definition of a Group 38
2. Subgroups 44
3. Isomorphisms 48
4. Homomorphisms 51
S. Equivalence Relations and Partitions 53
6. Cosets 57
7. Restriction of a Homomorphism to a Subgroup
8. Products of Groups 61
9. Modular Arithmetic 64
10. Quotient Groups 66
EXERCISES 69

59

vii

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Contents

viii

Chapter 3

78

Vector Spaces
1.
2.

Real Vector Spaces 78
Abstract Fields 82

3.
4.
5.

Bases and Dimension 87
Computation with Bases 94
Infinite-Dimensional Spaces 100

6. Direct Sums 102
EXERCISES 104

Chapter 4

1.

The Dimension Formula

2.

The Matrix of a Linear Transformation

109

3.

Linear Operators and Eigenvectors

4.

The Characteristic Polynomial

5.

Orthogonal Matrices and Rotations

6.

Diagonalization

7.

Systems of Differential Equations

115

120
123
133

138

155

Symmetry
1.
2.

Symmetry of Plane Figures 155
The Group of Motions of the Plane

3.

Finite Groups of Motions

4.

Discrete Groups of Motions

157

162
166

5.

Abstract Symmetry: Group Operations

6.
7.

The Operation on Cosets 178
The Counting Formula 180

8. Permutation Representations 182
9. Finite Subgroups of the Rotation Group
EXERCISES 188

Chapter 6

III

130

8. The Matrix Exponential
EXERCISES 145

Chapter 5

109

Linear Transformations

175

184

More Group Theory
1.
2.
3.

The Operations of a Group on Itself 197
The Class Equation of the Icosahedral Group
Operations on Subsets 203

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197
200


4.
5.
6.
7.

The Sylow Theorems 205

The Groups of Order 12 209
Computation in the Symmetric Group
The Free Group 217

211

8. Generators and Relations 219
9. The Todd-Coxeter Algorithm 223
EXERCISES 229

Chapter 7

Bilinear Forms
1.
2.
3.
4.
5.
6.
7.

Definition of Bilinear Form 237
Symmetric Forms: Orthogonality 243
The Geometry Associated to a Positive Form 247
Hermitian Forms 249
The Spectral Theorem 253
Conics and Quadrics 255
The Spectral Theorem for Normal Operators 259

8.

9.

Skew-Symmetric Forms 260
Summary of Results, in Matrix Notation

EXERCISES

Chapter 8

237

261

262

Linear Groups
1.

The Classical Linear Groups

270
270

2. The Special Unitary Group SU2 272
3. The Orthogonal Representation of SU2 276
4. The Special Linear Group SL2(lR) 281
5. One-Parameter Subgroups 283
6. The Lie Algebra 286
7. Translation in a Group 292
8. Simple Groups 295

EXERCISES 300

Chapter 9

Group Representations

307

1.
2.

Definition of a Group Representation 307
G-Invariant Forms and Unitary Representations

3.
4.

Compact Groups 312
G-Invariant Subspaces and Irreducible Representations

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310
314


Contents

x
5.

6.

Characters 316
Permutation Representations and the Regular
Representation 321

7.
8.

The Representations of the Icosahedral Group
One- Dimensional Representations 325

9.

Schur's Lemma, and Proof of the Orthogonality
Relations 325

10.

Representations of the Group SU2

EXERCISES

323

330

335

Chapter 10 Rings

345

1.

Definition of a Ring

2.
3.

Formal Construction of Integers and Polynomials
Homomorphisms and Ideals 353

4.

Quotient Rings and Relations in a Ring

5.

Adjunction of Elements

6.

Integral Domains and Fraction Fields

7.
8.

Maximal Ideals 370
Algebraic Geometry 373

EXERCISES

345
347

359

364
368

379

Chapter 11 Factorization

389

1.

Factorization of Integers and Polynomials

389

2.

Unique Factorization Domains, Principal Ideal Domains,
and Euclidean Domains 392

3.
4.

Gauss's Lemma 398
Explicit Factorization of Polynomials

402
406

5.

Primes in the Ring of Gauss Integers

6.
7.

Algebraic Integers 409
Factorization in Imaginary Quadratic Fields

8.

Ideal Factorization

9.

The Relation Between Prime Ideals of R and Prime
Integers 424
Ideal Classes in Imaginary Quadratic Fields 425
Real Quadratic Fields 433

10.
11.

414

419

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Contents

xi

12. Some Diophantine Equations
EXERCISES 440

437

Chapter 12 Modules

450

1. The Definition of a Module 450
2. Matrices, Free Modules, and Bases 452
3. The Principle of Permanence of Identities 456
4. Diagonalization of Integer Matrices 457
5. Generators and Relations for Modules 464
6. The Structure Theorem for Abelian Groups 471
7. Application to Linear Operators 476
8. Free Modules over Polynomial Rings 482
EXERCISES 483

Chapter 13 Fields

492

1. Examples of Fields 492
2. Algebraic and Transcendental Elements 493
3. The Degree of a Field Extension 496
4. Constructions with Ruler and Compass 500
5. Symbolic Adjunction of Roots 506
6. Finite Fields 509
7. Function Fields 515
8. Transcendental Extensions 525
9. Algebraically Closed Fields 527
EXERCISES 530

Chapter 14 Galois Theory
1. The Main Theorem of Galois Theory
2. Cubic Equations 543
3. Symmetric Functions 547
4. Primitive Elements 552
5. Proof of the Main Theorem 556
6. Quartic Equations 560
7. Kummer Extensions 565
8. Cyclotomic Extensions 567
9. Quintic Equations 570
EXERCISES 575

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537
537


Contents

xii

Appendix

585

Background Material
1.
2.

Set Theory 585
Techniques of Proof 589
3. Topology 593
4. The Implicit Function Theorem
EXERCISES 599

597

Notation

601

Suggestions for Further Reading

603

Index

607

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Preface
Important though the general concepts and propositions may be with which
the modern and industrious passion for axiomatizing and generalizing has
presented us, in algebra perhaps more than anywhere else, nevertheless I am
convinced that the special problems in all their complexity constitute the
stock and core of mathematics, and that to master their difficulties requires
on the whole the harder labor.
Herman Weyl

This book began about 20 years ago in the form of supplementary notes for my algebra classes. I wanted to discuss some concrete topics such as symmetry, linear
groups, and quadratic number fields in more detail than the text provided, and to
shift the emphasis in group theory from permutation groups to matrix groups. Lattices, another recurring theme, appeared spontaneously. My hope was that the concrete material would interest the students and that it would make the abstractions
more understandable, in short, that they could get farther by learning both at the
same time. This worked pretty well. It took me quite a while to decide what I
wanted to put in, but I gradually handed out more notes and eventually began teaching from them without another text. This method produced a book which is, I think,
somewhat different from existing ones. However, the problems I encountered while
fitting the parts together caused me many headaches, so I can't recommend starting
this way.
The main novel feature of the book is its increased emphasis on special topics.
They tended to expand each time the sections were rewritten, because I noticed over
the years that, with concrete mathematics in contrast to abstract concepts, students

often prefer more to less. As a result, the ones mentioned above have become major
parts of the book. There are also several unusual short subjects, such as the ToddCoxeter algorithm and the simplicity of PSL2 •

xiii

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xiv

Preface

In writing the book, I tried to follow these principles:
1. The main examples should precede the abstract definitions.
2. The book is not intended for a "service course," so technical points should be
presented only if they are needed in the book.
3. All topics discussed should be important for the average mathematician.
Though these principles may sound like motherhood and the flag, I found it useful to
have them enunciated, and to keep in mind that "Do it the way you were taught"
isn't one of them. They are, of course, violated here and there.
The table of contents gives a good idea of the subject matter, except that a first
glance may lead you to believe that the book contains all of the standard material in
a beginning algebra course, and more. Looking more closely, you will find that
things have been pared down here and there to make space for the special topics. I
used the above principles as a guide. Thus having the main examples in hand before
proceeding to the abstract material allowed some abstractions to be treated more
concisely. I was also able to shorten a few discussions by deferring them until the
students have already overcome their inherent conceptual difficulties. The discussion
of Peano's axioms in Chapter 10, for example, has been cut to two pages. Though
the treatment given there is very incomplete, my experience is that it suffices to give

the students the flavor of the axiomatic development of integer arithmetic. A more
extensive discussion would be required if it were placed earlier in the book, and the
time required for this wouldn't be well spent. Sometimes the exercise of deferring
material showed that it could be deferred forever-that it was not essential. This
happened with dual spaces and multilinear algebra, for example, which wound up on
the floor as a consequence of the second principle. With a few concepts, such as the
minimal polynomial, I ended up believing that their main purpose in introductory algebra books has been to provide a convenient source of exercises.
The chapters are organized following the order in which I usually teach a
course, with linear algebra, group theory, and geometry making up the first
semester. Rings are first introduced in Chapter 10, though that chapter is logically
independent of many earlier ones. I use this unusual arrangement because I want to
emphasize the connections of algebra with geometry at the start, and because, overall, the material in the first chapters is the most important for people in other fields.
The drawback is that arithmetic is given short shrift. This is made up for in the later
chapters, which have a strong arithmetic slant. Geometry is brought back from time
to time in these later chapters, in the guise of lattices, symmetry, and algebraic geometry.
Michael Artin
December 1990

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A Note for the Teacher

There are few prerequisites for this book. Students should be familiar with calculus,
the basic properties of the complex numbers, and mathematical induction. Some acquaintance with proofs is obviously useful, though less essential. The concepts from
topology, which are used in Chapter 8, should not be regarded as prerequisites. An
appendix is provided as a reference for some of these concepts; it is too brief to be
suitable as a text.
Don't try to cover the book in a one-year course unless your students have already had a semester of algebra, linear algebra for instance, and are mathematically
fairly mature. About a third of the material can be omitted without sacrificing much

of the book's flavor, and more can be left out if necessary. The following sections,
for example, would make a coherent course:
Chapter 1, Chapter 2, Chapter 3: 1-4, Chapter 4, Chapter 5: 1-7,
Chapter 6: 1,2, Chapter 7: 1-6, Chapter 8: 1-3,5, Chapter 10: 1-7,
Chapter 11: 1-8, Chapter 12: 1-7, Chapter 13: 1-6.
This selection includes some of the interesting special topics: symmetry of plane
figures, the geometry of SU2 , and the arithmetic of imaginary quadratic number
fields. If you don't want to discuss such topics, then this is not the book for you.
It would be easy to spend an entire semester on the first four chapters, but this
would defeat the purpose of the book. Since the real fun starts with Chapter 5, it is
important to move along. If you plan to follow the chapters in order, try to get to
that chapter as soon as is practicable, so that it can be done at a leisurely pace. It will
help to keep attention focussed on the concrete examples. This is especially imporxv

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A Note for the Teacher

xvi

tant in the beginning for the students who come to the course without a clear idea of
what constitutes a proof.
Chapter 1, matrix operations, isn't as exciting as some of the later ones, so it
should be covered fairly quickly. I begin with it because I want to emphasize the
general linear group at the start, instead of following the more customary practice of
basing examples on the symmetric group. The reason for this decision is Principle 3
of the preface: The general linear group is more important.
Here are some suggestions for Chapter 2:
1. Treat the abstract material with a light touch. You can have another go at it in

Chapters 5 and 6.
2. For examples, concentrate on matrix groups. Mention permutation groups only in
passing. Because of their inherent notational difficulties, examples from symmetry such as the dihedral groups are best deferred to Chapter 5.
3. Don't spend too much time on arithmetic. Its natural place in this book is Chapters 10 and 11.
4. Deemphasize the quotient group construction.
Quotient groups present a pedagogical problem. While their construction is conceptually difficult, the quotient is readily presented as the image of a homomorphism in
most elementary examples, and so it does not require an abstract definition. Modular
arithmetic is about the only convincing example for which this is not the case. And
since the integers modulo n form a ring, modular arithmetic isn't the ideal motivating example for quotients of groups. The first serious use of quotient groups comes
when generators and relations are discussed in Chapter 6, and I deferred the treatment of quotients to that point in early drafts of the book. But fearing the outrage of
the algebra community I ended up moving it to Chapter 2. Anyhow, if you don't
plan to discuss generators and relations for groups in your course, then you can defer
an in-depth treatment of quotients to Chapter 10, ring theory, where they play a
central role, and where modular arithmetic becomes a prime motivating example.
In Chapter 3, vector spaces, I've tried to set up the computations with bases in
such a way that the students won't have trouble keeping the indices straight. I've
probably failed, but since the notation is used throughout the book, it may be advisable to adopt it.
The applications of linear operators to rotations and linear differential equations in Chapter 4 should be discussed because they are used later on, but the temptation to give differential equations their due has to be resisted. This heresy will be
forgiven because you are teaching an algebra course.
There is a gradual rise in the level of sophistication which is assumed of the
reader throughout the first chapters, and a jump which I've been unable to eliminate
occurs in Chapter 5. Had it not been for this jump, I would have moved symmetry
closer to the beginning of the book. Keep in mind that symmetry is a difficult concept. It is easy to get carried away by the material and to leave the students behind.

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A Note for the Teacher

xvii

Except for its first two sections, Chapter 6 contains optional material. The last
section on the Todd-Coxeter algorithm isn't standard; it is included to justify the
discussion of generators and relations, which is pretty useless without it.
There is nothing unusual in the chapter on bilinear forms, Chapter 7. I haven't
overcome the main problem with this material, that there are too many variations on
the same theme, but have tried to keep the discussion short by concentrating on the
real and complex cases.
In the chapter on linear groups, Chapter 8, plan to spend time on the geometry
of 5U2 • My students complained every year about this chapter until I expanded the
sections on 5U2 , after which they began asking for supplementary reading, wanting
to learn more. Many of our students are not familiar with the concepts from topology when they take the course, and so these concepts require a light touch. But I've
found that the problems caused by the students' lack of familiarity can be managed.
Indeed, this is a good place for them to get an idea of what a manifold is. Unfortunately, I don't know a really satisfactory reference for further reading.
Chapter 9 on group representations is optional. I resisted including this topic
for a number of years, on the grounds that it is too hard. But students often request
it, and I kept asking myself: If the chemists can teach it, why can't we? Eventually
the internal logic of the book won out and group representations went in. As a dividend, hermitian forms got an application.
The unusual topic in Chapter 11 is the arithmetic of quadratic number fields.
You may find the discussion too long for a general algebra course. With this possibility in mind, I've arranged the material so that the end of Section 8, ideal factorization, is a natural stopping point.
It seems to me that one should at least mention the most important examples of
fields in a beginning algebra course, so I put a discussion of function fields into
Chapter 13.
There is always the question of whether or not Galois theory should be presented in an undergraduate course. It doesn't have quite the universal applicability
of most of the subjects in the book. But since Galois theory is a natural culmination
of the discussion of symmetry, it belongs here as an optional topic. I usually spend at
least some time on Chapter 14.
I considered grading the exercises for difficulty, but found that I couldn't do it
consistently. So I've only gone so far as to mark some of the harder ones with an
asterisk. I believe that there are enough challenging problems, but of course one always needs more of the interesting, easier ones.

Though I've taught algebra for many years, several aspects of this book are experimental, and I would be very grateful for critical comments and suggestions from
the people who use it.
"One, two, three, five, four ... "
"No Daddy, it's one, two, three, four, five."
"Well if I want to say one, two, three, five, four,
why can't I?"
"That's not how it goes."

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xviii

Acknowledgments
Mainly, I want to thank the students who have been in my classes over the years for
making them so exciting. Many of you will recognize your own contributions, and I
hope that you will forgive me for not naming you individually.
Several people have used my notes in classes and made valuable suggestionsJay Goldman, Steve Kleiman, Richard Schafer, and Joe Silverman among them.
Harold Stark helped me with the number theory, and Gil Strang with the linear algebra. Also, the following people read the manuscript and commented on it: Ellen
Kirkman, Al Levine, Barbara Peskin, and John Tate. I want to thank Barbara Peskin
especially for reading the whole thing twice during the final year.
The figures which needed mathematical precision were made on the computer
by George Fann and Bill Schelter. I could not have done them by myself.
Many thanks also to Marge Zabierek, who retyped the manuscript annually for
about eight years before it was put onto the computer where I could do the revisions
myself, and to Mary Roybal for her careful and expert job of editing the manuscript.
I've not consulted other books very much while writing this one, but the classics by Birkhoff and MacLane and by van der Waerden from which I learned the subject influenced me a great deal, as did Herstein's book, which I used as a text for
many years. I also found some good ideas for exercises in the books by Noble and
by Paley and Weichsel.
Some quotations, often out of context, are scattered about the text. I learned

the Leibnitz and Russell quotes which end Chapters 5 and 6 from V. I. Arnold, and
the Weyl quote which begins Chapter 8 is from Morris Klein's book Mathematical
Thoughtfrom Ancient to Modern Times.

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Chapter 1

Matrix Operations
~r~lid) mirtl a[[e~ M~jcnige cine C5ri)~e gcncnnt,
elner Q)crme~rung ooer elner Q)mnintlerung fa~ig i~,
otlCt mOb" lid) nod) ctm\\~ ~inb"lcl}CI\ otlCt Moon mcgnc~mCl\ {ii~t.
mdd)e~

Leonhard Euler

Matrices playa central role in this book. They form an important part of the theory,
and many concrete examples are based on them. Therefore it is essential to develop
facility in matrix manipulation. Since matrices pervade much of mathematics, the
techniques needed here are sure to be useful elsewhere.
The concepts which require practice to handle are matrix multiplication and
determinants.

1. THE BASIC OPERATIONS
Let m, n be positive integers. An m x n matrix is a collection of mn numbers arranged in a rectangular array:
n columns
a ll

(1.1)

m rows

:

[
amI

For example,

[~

3

~] is a 2 x 3 matrix.

The numbers in a matrix are called the matrix entries and are denoted by aij,
where i, j are indices (integers) with 1 ::; i ::; m and 1 ::; j ::; n. The index i is
called the row index, and j is the column index. So aij is the entry which appears in
1

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2

Matrix Operation

Chapter 1

the ith row and jth column of the matrix:
j

In the example above, all = 2, al3 = 0, and aZ3 = 5.
We usually introduce a symbol such as A to denote a matrix, or we may write it
as (aij).
A 1 x n matrix is called an n-dimensional row vector. We will drop the index i
when m = 1 and write a row vector as

(1.2)
The commas in this row vector are optional. Similarly, an m
dimensional column vector:

X

1 matrix is an m-

(1.3)

A 1 x 1 matrix [a] contains a single number, and we do not distinguish such a matrix from its entry.

(1.4) Addition of matrices is vector addition:
(aij) + (bij) = (Sij) ,
where sij = aij + bij for all i, j. Thus

[21 3150]+ 4[1 - 3°
The sum of two matrices A, B is defined only when they are both of the same
shape, that is, when they are m X n matrices with the same m and n.

(1.5) Scalar multiplication of a matrix by a number is defined as with vectors. The

result of multiplying a number c and a matrix (aij) is another matrix:
c(aij) = (bij),
where bij = cal} for all i, j. Thus

Numbers will also be referred to as scalars.

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Section 1

The Basic Operations

3

The complicated notion is that of matrix multiplication. The first case to learn
is the product AB of a row vector A (1.2) and a column vector B (1.3) which is
defined when both are the same size, that is, m = n. Then the product AB is the
1 x 1 matrix or scalar
(1.6)
(This product is often called the "dot product" of the two vectors.) Thus

[3

2{-

iJ ~

3 . J + J • (-I) + 2 . 4

~ 10.

The usefulness of this definition becomes apparent when we regard A and B as vectors which represent indexed quantities. For example, consider a candy bar containing m ingredients. Let a, denote the number of grams of (ingredient), per candy bar,
and let b, denote the cost of (ingredient), per gram. Then the matrix product AB = c
computes the cost per candy bar:
(grams/bar) . (cost/gram)

= (cost/bar).

On the other hand, the fact that we consider this to be the product of a row by a
column is an arbitrary choice.
In general, the product of two matrices A and B is defined if the number of
columns of A is equal to the number of rows of B, say if A is an x m matrix and B
is an m X n matrix. In this case, the product is an x n matrix. Symbolically,
(e x m) . (m X n) = (e x n). The entries of the product matrix are computed by
multiplying all rows of A by all columns of B, using rule (1.6) above. Thus if we denote the product AB by P, then

e

e

(1. 7)

This is the product of the ith row of A and the jth column of B.
j

. Pi.1··

ail' • • • • • • • • • aim

bmj

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Matrix Operation

4

Chapter 1

For example,
(1.8)
This definition of matrix multiplication has turned out to provide a very convenient
computational tool.
Going back to our candy bar example, suppose that there are candy bars.
Then we may form a matrix A whose ith row measures the ingredients of (bar)i. If
the cost is to be computed each year for n years, we may form a matrix B whose jth
column measures the cost of the ingredients in (year)j. The matrix product AB = P
computes the cost per bar: pij = cost of (bar); in (year)j.
Matrix notation was introduced in the nineteenth century to provide a shorthand way of writing linear equations. The system of equations

e

a"x,
aZ'x\

+ ... + a'nXn = b,
+ + aZnXn = hi

can be written in matrix notation as
(1.9)

AX = B,

where A denotes the coefficient matrix (au), X and B are column vectors, and AX is
the matrix product

D[]} r:J
Thus the matrix equation

represents the following system of two equations in three unknowns:
-Xz

+

2X3

=

2

+ 4xz - 6X3 = 1.
solution: x, = 1, Xz = 4, X3 =
3x,

Equation (1. 8) exhibits one
3.
Formula (1.7) defining the product can also be written in "sigma" notation as
m

r»

=

2: aikbkj = 2:k aub».
k=\

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Section 1

The Basic Operations

5

Each of these expressions is a shorthand notation for the sum (1.7) which defines the
product matrix.
Our two most important notations for handling sets of numbers are the L or
sum notation as used above and matrix notation. The L notation is actually the more
versatile of the two, but because matrices are much more compact we will use them
whenever possible. One of our tasks in later chapters will be to translate complicated
mathematical structures into matrix notation in order to be able to work with them
conveniently.
Various identities are satisfied by the matrix operations, such as the distributive
laws
(1.10)

A(B

+ B')

= AB

+ AB',

and

(A

+ A')B

= AB

+ A'B

and the associative law
(l.I1)

(AB)C = A(BC).

These laws hold whenever the matrices involved have suitable sizes, so that the
products are defined. For the associative law, for example, the sizes should be
A =
x m, B = m X nand, C = n X p, for some m, n, p. Since the two products
(1.11) are equal, the parentheses are not required, and we will denote them by ABC.
The triple product ABC is then an x p matrix. For example, the two ways of computing the product

e

e,

e

are

(AB)C =

[21 °° 21][2~ 0]~ [36 21]
=

and A(BC) =

[~][2

1] =

[~ ~l

Scalar multiplication is compatible with matrix multiplication in the obvious
sense:

(l.12)

c(AB) = (cA)B = A(cB).

The proofs of these identities are straightforward and not very interesting.
In contrast, the commutative law does not hold for matrix multiplication; that
is,

AB =F BA, usually.

(1.13)

e

e

In fact, if A is an x m matrix and B is an m X matrix, so that AB and BA are both
defined, then AB is x while BA is m x m. Even if both matrices are square, say
m X m, the two products tend to be different. For instance,

e e

1] [0 0]
[°0 °1][0° 0]1 = [0° °1] ,whIle. [0° 0][0
1 ° ° = ° °.
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Matrix Operation

6

Chapter 1

Since matrix multiplication is not commutative, care must be taken when
working with matrix equations. We can multiply both sides of an equation B = C on
the left by a matrix A, to conclude that AB = AC, provided that the products are
defined. Similarly, if the products are defined, then we can conclude that BA = CA.

We can not derive AB = CA from B = C!
Any matrix all of whose entries are is called a zero matrix and is denoted by
0, though its size is arbitrary. Maybe Ornxn would be better.
The entries aii of a matrix A are called its diagonal entries, and a matrix A
is called a diagonal matrix if its only nonzero entries are diagonal entries.
The square n X n matrix whose only nonzero entries are 1 in each diagonal position,

°

(1.14)

is called the n x n identity matrix. It behaves like 1 in multiplication: If A is an
m X n matrix, then
IrnA = A

and

At; = A.

Here are some shorthand ways of drawing the matrix In:

We often indicate that a whole region in a matrix consists of zeros by leaving it
blank or by putting in a single 0.
We will use * to indicate an arbitrary undetermined entry of a matrix. Thus

may denote a square matrix whose entries below the diagonal are 0, the other entries
being undetermined. Such a matrix is called an upper triangular matrix.
Let A be a (square) n X n matrix. If there is a matrix B such that
(1.15)

AB = In

and

BA = In,

then B is called an inverse of A and is denoted by A -I:
(1.16)
When A has an inverse, it is said to be an invertible matrix. For example, the matrix
A

=

2
[5

. A- 1
31]..
IS inverti'bl e. I
ts 'Inverse IS

=

[3
. seen b y computing
.
-5 -1]
2 ' as IS

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Section 1

The Basic Operations

the products

AA- I

7

and A-IA. Two more examples are:

We will see later that A is invertible if there is a matrix B such that either one of
the two relations AB = In or BA = In holds, and that B is then the inverse [see
(2.23)]. But since multiplication of matrices is not commutative, this fact is not obvious. It fails for matrices which aren't square. For example, let A = [1 2] and let
B

=

[~]. ThenAB = [1] = II, butBA = [~ ~]

=1=

/z.

On the other hand, an inverse is unique if it exists at all. In other words, there
can be only one inverse. Let B,B' be two matrices satisfying (1.15), for the same
matrix A. We need only know that AB = In (B is a right inverse) and that B'A = In

(B' is a left inverse). By the associative law, B'(AB) = (B'A)B. Thus
(1.17)
and so B'

B'

= B.

=

B'I

=

B'(AB)

=

(D'A)B

=

ID

=

D,

0

(1.18) Proposition.
product AB, and

Let

A, B

be n x n matrices. If both are invertible, so is their

More generally, if Al , ... , Am are invertible, then so is the product
inverse is A m- I "·A I- I.
I
1
1
1
Thus the inverse of [ I 2] [
]
[
is [ -

~

~]

Al ... Am,

and its

n[ !] [ -t].
1

Proof. Assume that A, B are invertible. Then we check that B- 1A -I is the inverse of AB:

and similarly

The last assertion is proved by induction on m [see Appendix (2.3)]. When m = 1,
the assertion is that if Al is invertible then A I- I is the inverse of At, which is trivial.
Next we assume that the assertion is true for m = k, and we proceed to check it for
m = k + 1. We suppose that AI, ... ,Ak+1 are invertible n x n matrices, and we denote by P the product AI ... Ak of the first k matrices. By the induction hypothesis, P
is invertible, and its inverse is Ak-I .•. Al -I. Also, Ak+1 is invertible. So, by what has
been shown for two invertible matrices, the product PAk+1 = AI··· AkAk+1 is invertible, and its inverse is Ak+I-lr l = Ak+I-IAk- 1 ···Al- I. This shows that the assertion is
true for m = k + 1, which completes the induction proof. 0

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8

Matrix Operation

Chapter 1

Though this isn't clear from the definition of matrix multiplication, we will see
that most square matrices are invertible. But finding the inverse explicitly is not a
simple problem when the matrix is large.
The set of all invertible n x n matrices is called the n-dimensional general linear group and is denoted by GL n . The general linear groups will be among our most
important examples when we study the basic concept of a group in the next chapter.
Various tricks simplify matrix multiplication in favorable cases. Block multiplication is one of them. Let M, M' be m x nand n x p matrices, and let r be an integer
less than n. We may decompose the two matrices into blocks as follows:
M = [A IB]

[~:J.

and M' =

where A has r columns and A' has r rows. Then the matrix product can be computed
as follows:
(1.19)

MM' = AA'

+ BB'.

This decomposition of the product follows directly from the definition of multiplication, and it may facilitate computation. For example,

Note that formula (1.19) looks the same as rule (1.6) for multiplying a row
vector and a column vector.
We may also multiply matrices divided into more blocks. For our purposes, a
decomposition into four blocks will be the most useful. In this case the rule for block
multiplication is the same as for multiplication of 2 x 2 matrices. Let r + s = nand
let k + = m. Suppose we decompose an m X n matrix M and an n X p matrix M'
into submatrices

e

M

=

[*J.

M'

[mJ.

where the number of columns of A is equal to the number of rows of A'. Then the
rule for block multiplication is
(1.20)

~] [~]
C'TLJ'
[ClD

=

[AA'
+ BC' AB' + BD']
-C-A-=-'- +-D-C-'-+---:-----:CB' + DD' .

For example,

~] .
[48T7O

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Section 2

Row Reduction

9

In this product, the upper left block is

[1

O][~ ~] + [5] [0

1] = [2

8], etc.

Again, this rule can be verified directly from the definition of matrix multiplication. In general, block multiplication can be used whenever two matrices are decomposed into submatrices in such a way that the necessary products are defined.
Besides facilitating computations, block multiplication is a useful tool for proving facts about matrices by induction.

1. ROW REDUCTION
Let A = (aij) be an m X n matrix, and consider a variable n x p matrix X = (Xij).
Then the matrix equation
(2.1)

Y = AX

defines the m X p matrix Y = (Yij) as a function of X. This operation is called left
multiplication by A:
(2.2)
Notice that in formula (2.2) the entry Yij depends only on XIj, .•. , xnj, that is, on the
jth column of X and on the ith row of the matrix A. Thus A operates separately on
each column of X, and we can understand the way A operates by considering its action on column vectors:

Left multiplication by A on column vectors can be thought of as a function
from the space of n-dimensional column vectors X to the space of m-dimensional
column vectors Y, or a collection of m functions of n variables:

(i = l, ... ,m).
It is called a linear transformation, because the functions are homogeneous and linear. (A linear function of a set of variables u\ ,... , u« is one of the form a, Uj +
... + akUk + c, where a" ... , ak,C are scalars. Such a function is homogeneous linear if the constant term c is zero.)
A picture of the operation of the 2 x 2 matrix
2-space to 2-space;

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[i ;]

is shown below. It maps