HM 802 February 8 , 2013 1 Preamble

The notes below will serve as the basis for the module lectures to be given on the topics of Jordan form, nonnegative matrices, and Markov chains. As you'll see, most of the proofs have been omitted in the notes. The intention is to sketch the proofs during the lectures, but also to have the students work through some of the details (using references such as those listed at the end of this document) as part of the self–study component of module. If you're having trouble getting ahold of some of the reference texts, I have copies of all of them in my office which you're welcome to borrow from time to time. I have also included some sample exercises. These are not intended to be terribly difficult, but if you want to discuss the problems with me once the module is over, feel free to do so. Suppose that A is a complex matrix of order n. A scalar λ ∈ C is an eigenvalue for A if there is a nonzero vector v ∈ C n such that Av = λv. In that case, the vector v is an eigenvector for A. Occasionally we refer to such a v as a right eigenvector for A. Note that there is a corresponding notion of a left eigenvector as well: a nonzero vector w ∈ C n is a left eigenvector for A corresponding to the eigenvalue λ provided that w * A = λw *. The spectral radius of a square matrix A, denoted ρ(A) is given by ρ(A) = {|λ||λ is an eigenvalue of A}. Suppose that the square matrix A has λ as an eigenvalue. A Jordan chain for λ is a finite list of nonzero vectors v 1 ,. .. , v k such that Av 1 = λv 1 , and for j =