Page 182 - Advanced Linear Algebra
P. 182
166 Advanced Linear Algebra
monic polynomial ²%³ of smallest degree for which ² ³ ~ . This
definition is equivalent to our definition.
The concept of minimal polynomial is also defined for matrices. The minimal
polynomial ²%³ of matrix ( C ²-³ is defined as the minimal polynomial
A
(
of the multiplication operator ( . Equivalently, ²%³ is the unique monic
polynomial ²%³ -´%µ of smallest degree for which ²(³ ~ .
Theorem 7.4
=
1 If ) are similar linear operators on , then ² % ³ ~ ² % ³ . Thus, the
minimal polynomial is an invariant under similarity of operators.
)
2 If ( ) are similar matrices, then ²%³ ~ ²%³ . Thus, the minimal
)
(
polynomial is an invariant under similarity of matrices.
)
3 The minimal polynomial of B ²= ³ is the same as the minimal
polynomial of any matrix that represents .
Cyclic Submodules and Cyclic Subspaces
:
Let us now look at the cyclic submodules of =
ºº#»»~-´%µ#~¸ ² ³²#³ ²%³-´%µ¹
=
which are -invariant subspaces of . Let ² % ³ be the minimal polynomial of
O ºº#»» and suppose that deg ² ²%³³ ~ . If ²%³# ºº#»», then writing
²%³ ~ ²%³ ²%³ b ²%³
where deg ²%³ deg ²%³ gives
²%³# ~ ´ ²%³ ²%³ b ²%³µ# ~ ²%³#
and so
ºº#»» ~ ¸ ²%³# deg ²%³ ¹
Hence, the set
8 c #¹ ~ ¸#Á #Á Ã Á c #¹
~ ¸#Á %#Á Ã Á %
spans the vector space ºº#»» . To see that is a basis for ºº#»» , note that any linear
8
combination of the vectors in has the form ²%³# for deg ² ²%³³ and so is
8
equal to if and only if ²%³ ~ . Thus, is an ordered basis for ºº#»» .
8
:
:
=
B
Definition Let ²= ³ . A -invariant subspace of is -cyclic if has a
basis of the form
8 ~ ¸#Á #Á Ã Á c #¹
=
8
for some #= and . The basis is called a -cyclic basis for .
