Math 721
Spring 2001
Homework 4
a)
Show that if is an invertible matrix, then
b)
All eigenvalues of are 0 if and only if
is nilpotent.
c)
If is nilpotent, then
the trace of
is 0.
d)
(A partial converse to b). Assume that is a real matrix (
with
) such that the trace of both
and
are 0. Show that
is nilpotent. What if
?
3. Find the rational canonical form, primary rational canonical form, and the Jordan canonical form for the following matrix: