Math 818: Problem set 11, due April 15

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, April 15, 2005. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
  1. Similarity (from the January 2002 qualifying exam):
  2. Jordan canonical form (from the January 2002 qualifying exam): Let A be the matrix
    [-1  -9  0  0]
    [ 1   5  0  0]
    [ 2   7  2  0]
    [ 4  13  0  2]
  3. Show that a square matrix with entries in a field k has the same invariant factors as does its transpose. Conclude that every square matrix is similar to its transpose.
  4. Give an example of a square matrix with rational entries which is similar to a diagonal matrix over the complexes but not over the rationals. Justify your answer.