Page 214 - Advanced Linear Algebra

198 Advanced Linear Algebra

We claim that the 's are the eigenvalues of and im²³ ~ ; . Theorem 2.25

implies that
=~ im ² ³ l Ä l im ² ³

# im
If ² ³ , then

²#³ ~ ² b Ä b ³# ~ ²#³

and so ; # . Hence, im ² ³ ‹ ; and so
= ~ im ² ³ lÄl im ² ³ ‹ ; lÄl ; ‹ =

and
which implies that im²³ ~ ;

=~ ; l Ä l ;

The converse also holds, for if =~ ; l Ä l ; and if is projection onto

along the direct sum of the other eigenspaces, then
;
bÄb ~
and since ~ , it follows that
~² b Ä b ³ ~ b Ä b
Theorem 8.12 A linear operator B ²= ³ is diagonalizable if and only if it
has a spectral resolution
~ b Ä
b
¹ is the spectrum of and

In this case, ¸Á à Á
ker
and ²³ ~ ;
im²³ ~ ;
£
Exercises
1. Let be the d matrix all of whose entries are equal to . Find the

1
minimal polynomial and characteristic polynomial of 1 and the
eigenvalues.
2. Prove that the eigenvalues of a matrix do not form a complete set of
invariants under similarity.

3. Show that ²= ³ is invertible if and only if is not an eigenvalue of .
B
4. Let be an d matrix over a field that contains all roots of the
-
(
characteristic polynomial of . Prove that det ² ( ³ is the product of the
(
eigenvalues of , counting multiplicity.
(

5. Show that if is an eigenvalue of , then ² ³ is an eigenvalue of ² ³ , for
any polynomial ²%³ . Also, if £ , then c is an eigenvalue for c .

B
6. An operator ²= ³ is nilpotent if ~ for some positive  . o