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Fearless Symmetry by Robert Gross, Avner Ash

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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:
£
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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