Page 88 - Advanced Linear Algebra

72 Advanced Linear Algebra

9
´µ ~ 4 8 Á 9
and so
c

´µ ~ ´ µ ´ µ ´ µ ~ ´ 9 c µ 9

9
9
9
from which it follows that and are similar. Conversely, suppose that and

are similar, say
~ c
where is an automorphism of . Suppose also that is represented by the

=
, that is,
matrix ( C

(~ ´ µ 8
8
for some ordered basis . Then ´µ ~ 4 9 Á 8 and so
8

8
8
´ µ ~ ´ 8 c µ ~ ´µ ´ µ ´µ c ~ 4 9 Á 8 ´ µ 4 9 c 8
8
8
8
Á
It follows that
c

8
( ~ ´ µ ~ 4 8 9 ´µ 4 89 ~ ´µ 9 8 Á
Á
and so also represents . By symmetry, we see that and are represented
(

by the same set of matrices. This completes the proof.
We can summarize the sitiation with respect to similarity in Figure 2.2. Each
J
I
B
J ²-³
similarity class in ²= ³ corresponds to a similarity class in C : is
I
the set of all matrices that represent any  I and is the set of all operators

in B that are represented by any 4  J²= ³ .
W V similarity classes

W V of L(V)
I
[W ] [V ] Similarity classes
B
B
[W ] [V ] of matrices
J C C
Figure 2.2
Invariant Subspaces and Reducing Pairs
:
The restriction of a linear operator B ²= ³ to a subspace of = is not
necessarily a linear operator on . This prompts the following definition.
: