MATRICES 115

also important in topology and geometry, not to mention physics,

chemistry, economics, and many other areas. For example, the

study of special relativity can be viewed as the theory of a certain

group of 4-by-4 matrices, the Lorentz group. Elementary quantum

mechanics involves representations of operators by matrices, some

of them of inﬁnite size. Economists use matrices in game theory.

And so on.

There are groups of matrices that provide us with a series of

standard examples of groups whose group law is not commutative

(i.e., x ◦ y is not necessarily equal to y ◦ x). This partially explains

why they are so useful. Many of our most important groups, includ-

ing SO(3) and the absolute Galois group G, have a noncommutative

group law. If we want to get a useful snapshot of such groups by

representing them inside a standard group, we need a large supply

of noncommutative standard groups. We get this supply from the

permutation and matrix groups.

A word about terminology: It gets boring using the word matrix

both as a noun, naming the entity we are soon to deﬁne, and as an

adjective. So we have a synonym for the adjectival use: linear.A

linear representation is just a matrix representation, and a linear

group is just a group of matrices. B

EWARE: The adjective “linear”

has many other meanings, such as “in the shape of a straight line,”

and it has other uses in mathematics and even in representation

theory. We hope these multiple meanings will not be a problem

in the explanations to come. We will usually use linear to mean

matricial.

Matrices and Their Entries

What is a matrix? In mathematics, the usual deﬁnition is “a

rectangular array of o bjects.” This is a correct deﬁnition, but it

seems too vague to be of much use. It conjures up armies of objects

marching past, and perhaps forming and reforming their arrays.

We try to understand this deﬁnition as concretely as possible .

First, we discuss the objects that are going to appear in the

matrices. We do not really want to consider just any old objects.

116 CHAPTER 10

It is true that technically this is a matrix:

£ ℵ

♥

But we do not want this kind of thing sneaking up into a theory, any

more than a platoon of soldiers. So we agree that all the objects in

our rectangular array will be elements of some number system R.

We call such a matrix an R-matrix. For example, we refer to a

Z-matrix, or a C-matrix, or an F

2

-matrix.

We call our matrix an r-by-c matrix if it has r rows and c columns.

Consider, for example, the particular number system F

2

, consisting

of the numbers 0 and 1. Some F

2

-matrices are the following:

1

,

10

,

110

,

0

0

,

⎡

⎢

⎣

1

0

1

⎤

⎥

⎦

,

1101

1011

EXERCISE: Draw all possible 2-by-3 F

2

-matrices. How many

are there? (You may need a fairly large sheet of paper for

this .)

If we want to specify a matrix, we can draw it or we can say which

numbers are where. For example, the F

2

matrix

100

010

can be

drawn just like that, or we can tell you that the ﬁrst row is

[

100

]

and the second row

[

010

]

, or we can tell you that the ﬁrst column

is

1

0

, the second is

0

1

, and the third is

0

0

. Or, most painstaking of

all, we can phone you the following instructions: Put a 1 in the ﬁrst

row and ﬁrst column, put a 0 in the ﬁrst row and second column,

and so on. Although this last method is painful, it does not matter if

we get the order of the instructions mixed up. It also tells you how

you can inquire about a speciﬁc element: “There was static on the

line and I didn’t hear what number was in the second row and ﬁrst

column. Oh, it was a 0? ...”

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