Page 190 - Advanced Linear Algebra
P. 190
174 Advanced Linear Algebra
;
since is linearly independent. Hence, ²%³ has smallest degree among all
(
polynomials satisfied by and so ² % ³ ~ ( ² % ³ . Finally,
deg² ²%³³ ~ deg² ²%³³ ~ deg² ²%³³
(
(
is cyclic with -cyclic basis ,
8
For part 2), we have already proved that if =
then ´ µ ~ *´ ²%³µ . For the converse, if ´ µ ~ *´ ²%³µ , then part 1) implies
8
8
that is nonderogatory. Hence, Theorem 7.11 implies that = is cyclic. It is
8
clear from the form of *´ ²%³µ that is a -cyclic basis for .
=
The Big Picture
If Á B²= ³ , then Theorem 7.2 and the fact that the elementary divisors form
a complete invariant for isomorphism imply that
¯ = = ¯ ElemDiv ² ³ ~ ElemDiv ² ³
Hence, the multiset of elementary divisors is a complete invariant for similarity
of operators. Of course, the same is true for matrices:
( ) ¯ - - ¯ ElemDiv ²(³ ~ ElemDiv ²)³
( )
where we write - in place of - .
( (
The connection between the elementary divisors of an operator and the
elementary divisors of the matrix representations of is described as follows. If
, then the coordinate map ¢ = - is also a module isomorphism
(~ ´ µ 8 8
¢= ¦ - . Specifically, we have
8 (
8 ² ² ³#³
8 ~ ´ ² ³#µ ~ ²´ µ ³´#µ ~ ²(³ 8 ²#³
8
8
and so 8 preserves -´%µ -scalar multiplication. Hence,
(~ ´ µ 8 8 for some ¬ = - (
For the converse, suppose that ( . If we define = by ¢ = - ~ ,
where is the th standard basis vector, then 8 ~ ² Á Ã Á ³ is an ordered
basis for and ~ 8 is the coordinate map for . Hence, 8 is a module
8
=
isomorphism and so
²#³ ~ 8 ( ² #³
8
for all #= , that is,
´#µ ~ 8 ( ²´#µ ³
8
which shows that (~ ´ µ 8 .
Theorem 7.13 Let = be a finite-dimensional vector space over - . Let
Á B²= ³ and let (Á ) C ²-³.
