  Krawtchouk matrices have as entries values of the Krawtchouk polynomials for
nonnegative integer arguments. We show how they arise as condensed
Sylvester-Hadamard matrices via a binary shuffling function. The underlying
symmetric tensor algebra is presented. Our approach is used to solve Kac'
formulation of the Ehrenfest urn model. Connections with quantum and classical
random walks are shown as well as various extensions of the classical
polynomials.
