Math 421/621

Spring 2000

Homework 4

 

  1. Is the matrix  equivalent to a diagonal matrix over the real numbers (how about to an upper triangular matrix)? What if we are over the complex numbers?

 

  1. Consider the matrix  

a)      Find the invariant factors and the elementary divisors of this matrix. What is the minimal polynomial of this matrix?

b)      Use the answer from part a) to find the rational canonical form (both for elementary divisors and invariant factors) of this matrix and the Jordan form for this matrix.

 

  1. Consider the matrix

a)      Find the rational canonical forms of this matrix by considering it as a real matrix.

b)      Now consider the matrix to be complex and find the Jordan canonical form.

 

  1. Show that the fields Q and Q are not isomorphic fields.

 

  1. Construct (if possible) a field of 4 elements.