Harbourne's Math 818 Problem set 11

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.)

Similarity (from the January 2002 qualifying exam):
- (a) Write down square matrices whose characteristic polynomial is (t² +1)³, such that every complex matrix with this characteristic polynomial is similar to one of those you've written down, and such that no two of the ones you've written down are similar to each other. (I.e., write down a representative from each similarity class of complex matrices with the given characteristic polynomial.)
- (b) Now write down a representative from each similarity class of rational matrices with the given characteristic polynomial.
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]
```
- (a) Show that the characteristic polynomial is (t - 2)⁴.
- (b) Find the Jordan canonical form J of A. Justify your answer.
- (c) We know J = P^-1AP. Find either P or P^-1 (your choice), but be sure to indicate which it is you've found.
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.
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.
- (a) Find all possible Jordan canonical forms (up to ordering of the blocks) for a matrix whose characteristic polynomial is (t-2)²(t-5)³.
- (b) Find all Jordan and rational canonical forms of matrices having characteristic polynomial (t-2)²(t-5)³ and whose space of eigenvectors with eigenvalue 2 is one-dimensional, while the space of eigenvectors with eigenvalue 5 is two-dimensional.