Page 251 - Advanced Linear Algebra
P. 251
Structure Theory for Normal Operators 235
and so
i
ker²³ ~ ker² ³
)
3 For any integer ,
ker²³ ~ ker² ³
)
4 The minimal polynomial ²%³ is a product of distinct prime monic
polynomials.
5)
#~ # ¯ i #~ #
) : ; = with relatively prime orders, then : . ;
6 If and are submodules of
) ; ; .
7 If and are distinct eigenvalues of , then
Proof. We leave part 1) for the reader. For part 2), normality implies that
º #Á $» ~ º i #Á #» ~ º i #Á #» ~ º #Á i i #»
We prove part 3) first for the operator ~ i , which is self-adjoint , that is,
i ~² i i ³ ~ i ~
If #~ for , then
~ º # Á c # » ~ º c # Á c # »
and so c #~ . Continuing in this way gives #~ . Now, if #~ for
, then
i
#~² i ³ #~² ³ #~
and so #~ . Hence,
i
~ º #Á#» ~ º #Á#» ~ º #Á #»
and so #~ .
For part 4 , suppose that
)
²%³ ~ ²%³ ²%³
where ²%³ is monic and prime. Then for any # = ,
² ³´ ² ³#µ ~
and since ² ³ is also normal, part 3) implies that
² ³´ ² ³#µ ~
for all #= . Hence, ² ³ ² ³~ , which implies that ~ . Thus, the prime
factors of ²%³ appear only to the first power.
