Page 252 - Advanced Linear Algebra

236 Advanced Linear Algebra

Part 5) follows from part 2):

i

ker²c ³ ~ ker´²c ³ µ ~ ker² i c ³

For part 6), if ²:³ ~ ²%³ and ²;³ ~ ²%³ , then there are polynomials ²%³
and ²%³ for which ²%³ ²%³ b ²%³ ²%³ ~ and so

²³ ²³ b ²³ ²³ ~

:
;
Now, ~ ² ³ ² ³ annihilates and ~ ² ³ ² ³ annihilates . Therefore
i
also annihilates and so;

º:Á ;» ~ º² b ³:Á ;» ~ º :Á ;» ~ º:Á i ;» ~ ¸ ¹

; ; are
Part 7) follows from part 6), since ² ³ ~ % c and ² ³ ~ % c
relatively prime when £ . Alternatively, for  # ; and  $ ; , we have
º#Á $»~º #Á $»~º#Á i $»~º#Á $»~ º#Á $»

and so £ implies that # º Á $ » ~ .
The Spectral Theorem for Normal Operators
Theorem 10.8 implies that when -~ d , the minimal polynomial ²%³ splits

into distinct linear factors and so Theorem 8.11 implies that is diagonalizable,

that is,
=~ ; l Ä l ;
Moreover, since distinct eigenspaces of a normal operator are orthogonal, we
have
=~ ; p Ä p ;
and so is unitarily diagonalizable.

The converse of this is also true. If = has an orthonormal basis E ~
¸# ÁÃÁ# ¹ of eigenvectors for , then since ´ µ and E ´ µ ~ ´ µ are i E i E

diagonal, these matrices commute and therefore so do and .
i

Theorem 10.9 (The spectral theorem for normal operators: complex case )
Let be a finite-dimensional complex inner product space and let  B ² = . ³
=
The following are equivalent:
)
1 is normal.
)
2 is unitarily diagonalizable, that is,
=~ ; p Ä p ;