Problem 1
A matrix
is called
nilpotent if
for some
. Show that if a matrix is nilpotent, then the only eigenvalue
of
is
.
Problem 2
Prove that the eigenvalues of an upper-triangular matrix are its diagonal entries.
Problem 3
Prove that a linear operator on an
-dimensional vector space which has
distinct eigenvalues is diagonalizable.
Problem 4
Prove that if
for
is invertible,
then so is
, and
. (You might
want to use this result in problem 6 on page 106).
Problem 5
Let
be the vector space of all continuous
real-valued functions on the real line. Prove that the operator
given by
has no eigenvalues.
Problem 6
a) Give a matrix (with real entries) which is diagonalizable over
,
but is not diagonalizable over
.
b) Give an example of a matrix which is not diagonalizable over
.
(
Hint: in both parts, it can be useful to think first about an operator
with such properties).