Math 721
Spring 2001
Homework 4
- Let
be an 
matrix. Show
that 
satisfies its
characteristic polynomial det(
) (where 
is the 
identity
matrix).
- An
matrix 
is said to be nilpotent
if 
for some 
Additionally the
trace of a matrix is the sum of its diagonal elements. Prove the following
(you may assume that the matrix has entries from a field).
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:
- Prove
that a
matrix,
over a field 
is similar to a diagonal matrix if and only if there is
a basis of 
consisting of eigenvectors of 