Page 183 - Advanced Linear Algebra

The Structure of a Linear Operator 167

Thus, a cyclic submodule ºº#»» of = with order ²%³ of degree is a -cyclic
subspace of of dimension . The converse is also true, for if
=

8 ~ ¸#Á #Á Ã Á c #¹

:
is a basis for a -invariant subspace of = , then is a submodule of = .
:
O

Moreover, the minimal polynomial of : has degree , since if
# ~ c #c #cÄc c c #
satisfies the polynomial
then O :
% c b %
²%³ ~ b %bÄb c

but none of smaller degree since is linearly independent.
8
=
Theorem 7.5 Let be a finite-dimenional vector space and let : ‹ = . The
following are equivalent:
=
1 : ) is a cyclic submodule of with order ² % ³ of degree

2 : ) is a -cyclic subspace of of dimension .

=
We will have more to say about cyclic modules a bit later in the chapter.
Summary
The following table summarizes the connection between the module concepts
and the vector space concepts that we have discussed so far.
-´%µ-Module = - -Vector Space =
Scalar multiplication: ²%³# Action of ² ³ ² ³²#³
:
Submodule of = -Invariant subspace of =
Annihilator: Annihilator:

ann²= ³ ~ ¸ ²%³ “ ²%³= ~ ¸ ¹¹ ann²= ³ ~ ¸ ²%³ “ ² ³²= ³ ~ ¸ ¹¹

Monic order ²%³ of = : Minimal polynomial of :
ann²= ³ ~ º ²%³» ²%³ has smallest deg with ² ³ ~

= :
=
Cyclic submodule of -cyclic subspace of :
ºº#»» ~ ¸ ²%³# “ deg ²%³  deg ²%³¹ º#Á #Á à Á c ²#³»Á ~ deg² ²%³³
The Primary Cyclic Decomposition of =
We are now ready to translate the cyclic decomposition theorem into the
.
language of =
Definition Let ²= ³ .
B

1 The elementary divisors and invariant factors of are the monic
)

.
elementary divisors and invariant factors, respectively, of the module =

We denote the multiset of elementary divisors of by ElemDiv²³ and the

multiset of invariant factors of by InvFact²³ .