Page 215 - Advanced Linear Algebra
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Eigenvalues and Eigenvectors 199
)
a Show that if is nilpotent, then the spectrum of is ¸ ¹ .
)
b Find a nonnilpotent operator with spectrum ¸ ¹ .
7. Show that if Á B²= ³ and one of and is invertible, then
and so and have the same eigenvalues, counting multiplicty.
8. ( Halmos)
)
a Find a linear operator that is not idempotent but for which
²c ³ ~ .
b Find a linear operator that is not idempotent but for which
)
²c ³ ~ .
)
c Prove that if ²c ³ ~ ²c ³ ~ , then is idempotent.
9. An involution is a linear operator for which ~ . If is idempotent
what can you say about c ? Construct a one-to-one correspondence
between the set of idempotents on and the set of involutions.
=
d
10. Let (Á ) 4 ² ³ and suppose that ( ~) ~0Á ()( ~) c but
(£ 0 and ) £ 0. Show that if * 4 ² ³ commutes with both ( and ),
d
then *~ 0 for some scalar d .
11. Let ²= ³ and let
B
:~ º#Á #Á Ã Á c #»
be a -cyclic submodule of = with minimal polynomial ²%³ where ²%³
is prime of degree . Let ~ ² ³ restricted to º # » . Show that is the
:
direct sum of -cyclic submodules each of dimension , that is,
:~ ; l Ä l ;
Hint: For each , consider the set
8 ~ ¸ #Á ² ³ #ÁÃÁ ² ³ c #»
12. Fix . Show that any complex matrix is similar to a matrix that looks
just like a Jordan matrix except that the entries that are equal to are
replaced by entries with value , where is any complex number. Thus, any
complex matrix is similar to a matrix that is “almost” diagonal. Hint :
consider the fact that
v y v y v y v y
c ~
w z w z w c z w z
13. Show that the Jordan canonical form is not very robust in the sense that a
small change in the entries of a matrix may result in a large jump in the
(
entries of the Jordan form . Hint : consider the matrix
1
(~ > ?
What happens to the Jordan form of ( as ¦ ?
