Page 87 - Advanced Linear Algebra

Linear Transformations 71

7~ 4 89Á . Hence
c
)~ 4 89Á ´ µ 4 89 ~ ´ µ 9

8
Á
Hence, also represents . By symmetry, we see that and represent the
)
(
)

same set of linear operators. This completes the proof.
We will devote much effort in Chapter 7 to finding a canonical form for
similarity.
Similarity of Operators
We can also define similarity of operators.

Definition Two linear operators  Á B = ² ³ are similar , denoted by — , if

there exists an automorphism ²= ³ for which
B
~ c
The equivalence classes associated with similarity are called similarity
classes.
8
9
Note that if ~² Á Ã Á ³ and ~² Á Ã Á ³ are ordered bases for , then
=

4 98Á ~ ²´ µ “Ä“´ µ ³

8
8

is an automorphism of and
=
Now, the map defined by ² ³ ~
4 98Á ~ ²´ ² ³µ “ Ä “ ´ ² ³µ ³ ~ ´ µ 8 8 8
Conversely, if is an automorphism and ~ ² Á ÃÁ ³ is an ordered
8¢= ¦ =

basis for , then ~ ² ~ ² ³ÁÃÁ ~ ² ³³ is also a basis:
=
9

´µ ~ ²´² ³µ “ Ä “ ´² ³µ ³ ~ 4 9 8 8 8 Á 8
The analog of Theorem 2.19 for linear operators is the following.

Theorem 2.20 Let = be a vector space of dimension . Then two linear
operators and on are similar if and only if there is a matrix (  C that
=

represents both operators, but with respect to possibly different ordered bases.
In this case, and are represented by exactly the same set of matrices in C .

Proof. If and are represented by ( C , that is, if

´ µ ~( ~´ µ 9 8
for ordered bases and , then
9
8

´µ ~ ´ µ ~ 4 9 8 8 ´ µ 4 8 9 Á Á 9
9
As remarked above, if is defined by ² ³ ~ , then
¢= ¦ =