Card Shuffling and the Polynomial Numerical Hull of Degree k

Anne Greenbaum

Mathematics Department, Box 354350
University of Washington
Seattle, WA 98195


Abstract

The behavior of a nonnormal matrix is not necessarily governed by its eigenvalues, so there is considerable interest in finding sets that one can associate with nonnormal matrices to give more information than the spectrum alone can provide about their behavior under the action of various functions. In this talk we consider one such type of set called the polynomial numerical hull of degree k. This is the set of all complex numbers z such that ||p(A)|| is greater than or equal to |p(z)| for all polynomials p of degree k or less. We discuss one particular application of these sets in explaining the cutoff phenomenon that is often observed in Markov processes; e.g., why it takes 7 riffle shuffles to randomize a deck of cards.