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: