Page 185 - Advanced Linear Algebra

The Structure of a Linear Operator 169

cyclic subspace ºº# »» has -cyclic basis

Á
8 Á Á ~# 2 Á Á # Á Ã Á c # 3 Á
Á
Á
and dim²ºº# »»³ ~ deg² ³ . Hence,
Á

dim²= ³ ~ deg² ³
Á

~
We will call the basis
H 8 ~ Á
Á
for the elementary divisor basis for .
=
=

(
Recall that if =~ ( l ) and if both and are -invariant subspaces of ,
=
)

the pair ²(Á )³ is said to reduce . In module language, the pair ²(Á )³ reduces
)
=
(
if and are submodules of and

=~ ( l )
We can now translate Theorem 6.15 into the current context.
B
Theorem 7.7 Let ²= ³ and let

=~ ( l )
)
1 The minimal polynomial of is

²%³ ~ lcm ² ²%³Á O O ( ) ²%³³

) is the direct sum of the primary
2 The primary cyclic decomposition of =
cyclic decompositons of ( and ) ; that is, if
( ~ ºº »» and ) ~ ºº »»
Á
Á
are the primary cyclic decompositions of ( and ) , respectively, then
= ~ 4 ºº »» l 4 Á 5 ºº »»5 Á
.
is the primary cyclic decomposition of =
)
3 The elementary divisors of are

ElemDiv²³ ~ ElemDiv²O ³ r ElemDiv²O ³

(
)
where the union is a multiset union; that is, we keep all duplicate
members.