chapter 3

14 Pages

## Powers of Matrices

In this chapter we consider powers of square matrices and describe them in terms of a digraph, different from the Ko¨nig digraph, that we associate with a square matrix. The basic result here is a theorem by which the entries of powers of a square matrix can be calculated by enumeration of certain walks in the associated digraph. As applications we consider Markov chains, finite automata, and counting permutations with certain restrictions. We also show how a certain structured matrix called a circulant results from the powers of a matrix whose digraph is a cycle.