Page 181 - Advanced Linear Algebra
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The Structure of a Linear Operator 165
²
²#³ ~ #³
and since is bijective, this is equivalent to
² c ³ # ~ #
that is, ~ c . Since a module isomorphism from = to = is a vector space
isomorphism as well, the result follows.
For the converse, suppose that is a vector space automorphism of = and
~ c , that is, ~ . Then
²% #³ ~ ² #³ ~ ² #³ ~ % ² #³
-
and the -linearity of implies that for any polynomial ² % ³ - ´ % , µ
² ² ³#³ ~ ² ³ #
Hence, is a module isomorphism from = to = .
Submodules and Invariant Subspaces
There is a simple connection between the submodules of the -´%µ -module =
and the subspaces of the vector space . Recall that a subspace of is -
=
=
:
invariant if : : .
:
Theorem 7.3 A subset : = is a submodule of = if and only if is a -
invariant subspace of .
=
Orders and the Minimal Polynomial
,
We have seen that the annihilator of =
ann²= ³ ~ ¸ ²%³ -´%µ ²%³= ~ ¸ ¹¹
is a nonzero principal ideal of -´%µ , say
ann²= ³ ~ º ²%³»
Since the elements of the base ring -´%µ of = are polynomials, for the first time
in our study of modules there is a logical choice among all scalars in a given
associate class: Each associate class contains exactly one monic polynomial.
Definition Let ²= ³ . The unique monic order of = is called the minimal
B
polynomial for and is denoted by ²%³ or min ² ³ . Thus,
ann²= ³ ~ º ²%³»
,
In treatments of linear algebra that do not emphasize the role of the module =
the minimal polynomial of a linear operator is simply defined as the unique
