Matrices permutation equivalent to primitive matrices

Abstract

We give a necessary and sufficient condition for an n× n (0,1) matrix (or more generally, an n× n nonnegative matrix) to be permutation equivalent to a primitive matrix. More precisely, except for two simple permutation equivalent classes of n× n (0,1) matrices, each n× n (0,1) matrix having no zero row or zero column is permutation equivalent to some primitive matrix. As an application, we use this result to characterize the subsemigroup of B n ( B n is the multiplicative semigroup of n× n Boolean matrices) generated by all the primitive matrices and permutation matrices. We also consider a more general problem and give a necessary and sufficient condition for an n× n nonnegative matrix to be permutation equivalent to an irreducible matrix with given imprimitive index.