Page 188 - Advanced Linear Algebra

172 Advanced Linear Algebra

Indecomposable Modules
, each prime factor
We have also seen (Theorem 6.19) that, in the language of =
²%³ of the minimal polynomial ²%³ gives rise to a cyclic submodule > of =

of prime order ²%³ .

B
Theorem 7.10 Let  ²= ³ and let ²%³ be a prime factor of ²%³ . Then =

has a cyclic submodule > of prime order ² % . ³
For a module of prime order, we have the following.

Theorem 7.11 For a module > of prime order ² % ³ , the following are
equivalent:
) is cyclic
1 >
) is indecomposable
2 >
)
3 ²%³ is irreducible

)
4 is nonderogatory, that is, ²%³ ~ ²%³

5)dim²> ³ ~ deg² ²%³³ .

Our translation project is now complete and we can begin to look at issues that
.
are specific to the modules =
Companion Matrices
via the matrix representations of
We can also characterize the cyclic modules =
the operator , which is obviously something that we could not do for arbitrary

modules. Let =~ ºº#»» be a cyclic module, with order

% c b %

²%³ ~ b %bÄb c
and ordered -cyclic basis

8 2 ~#Á #Á Ã Á c #3
Then

²#³ ~ b #
for   c and
² c # ³ ~ #
c
~ c² b c bÄb ³#
c
~c # c c # c Ä c #
and so