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
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.