Page 180 - Advanced Linear Algebra

164 Advanced Linear Algebra

This gives rise to two different directions for further study. First, we can search
for a characterization of those linear operators that can be represented by
diagonal matrices. Such operators are called diagonalizable . Second, we can
search for a different type of “simple” matrix that does provide a set of
canonical forms for similarity. We will pursue both of these directions.
The Module Associated with a Linear Operator

=
-

B
If ²= ³ , we will think of not only as a vector space over a field but
also as a module over -´%µ , with scalar multiplication defined by
²%³# ~ ² ³²#³

We will write = to indicate the dependence on . Thus, = and = are modules
with the same ring of scalars -´%µ , although with different scalar multiplication
if . £
.
Our plan is to interpret the concepts of the previous chapter for the module =
B
First, if dim²= ³ ~ , then dim² ²= ³³ ~ . This implies that = is a torsion

module. In fact, the b vectors

ÁÁ
Á Ã Á
are linearly dependent in B , which implies that ²= ³ ² ³ ~ for some nonzero
polynomial ²%³  -´%µ . Hence, ²%³  ann ²= ³ and so ann ²= ³ is a nonzero

principal ideal of -´%µ .
Also, since is finitely generated as a vector space, it is, a fortiori, finitely
=
generated as an -´%µ -module. Thus, is a finitely generated torsion module
=
over a principal ideal domain -´%µ and so we may apply the decomposition
theorems of the previous chapter. In the first part of this chapter, we embark on
a “translation project” to translate the powerful results of the previous chapter
.
into the language of the modules =
Let us first characterize when two modules = and = are isomorphic.

Theorem 7.2 If  Á B = ² ³ , then

=š = ¯ —
In particular, is a module isomorphism if and only if is a vector
¢= ¦ =
space automorphism of satisfying
=
~ c

Proof. Suppose that ¢= ¦ = is a module isomorphism. Then for #  = ,
²%#³ ~ %² #³
which is equivalent to