Page 217 - Advanced Linear Algebra

Eigenvalues and Eigenvectors 201

)
a If all the roots of the characteristic polynomial of lie in prove that
(
-
)
( ( is similar to its transpose ! . Hint: Let be the matrix
v Ä y
Å x Ç {
)~ x {
ÇÇ Å
w Ä z
with 's on the diagonal that moves up from left to right and 's

elsewhere. Let be a Jordan block of the same size as . Show that
)
1
!
)1) c ~ 1 .
)
b Let (Á )  C ²-³ . Let be a field containing . Show that if and
-
(
2
) 2 are similar over , that is, if ) ~ 7 ( 7 c 7  C ² where 2 ³ , then
-
( ) and are also similar over , that is, there exists 8  ² -C ³ for
which )~ 8(8 c .
c Show that any matrix is similar to its transpose.
)
The Trace of a Matrix
18. Let ( C ²-³ . Verify the following statements.
a) tr² A³ ~ tr²(³ , for  - .
b) tr²( b )³ ~ tr²(³ b tr²)³ .
c) tr²()³ ~ tr²)(³ .
d) tr²()*³ ~ tr²*()³ ~ tr²)*(³ . Find an example to show that
tr²()*³ may not equal tr²(*)³.
e) The trace is an invariant under similarity.
f) If - is algebraically closed, then the trace of is the sum of the
(
eigenvalues of .
(
19. Use the concept of the trace of a matrix, as defined in the previous exercise,
to prove that there are no matrices , ()  Cd for which
² ³

() c )( ~ 0
20. Let ;¢ C ²-³ ¦ - be a function with the following properties. For all
matrices (Á )  C ²-³ and  - ,
)
1 ;² A³ ~ ;²(³
)
2 ;²( b )³ ~ ;²(³ b ;²)³
)
3 ;²()³ ~ ;²)(³
Show that there exists - for which ;²(³~ tr²(³ , for all
²-³.
( C
Commuting Operators
Let
< ~¸  ²= ? ³ “  ¹
B
be a family of operators on a vector space . Then is a commuting family if
<
=
every pair of operators commutes, that is, ~ for all  Á < . A subspace