Math 721

Spring 2001

Homework 4

 

  1. Let  be an  matrix. Show that  satisfies its characteristic polynomial det() (where  is the  identity matrix).

 

  1. 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:

 

 

  1. 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