Irreducible matrices †

Abstract

Frobenius proved that if Ais an irreducible matrix, then there exists a permutation matrix Psuch that PAP^T is equal to a partitioned matrix (A_ij)whose main diagonal blocks A₁₁ ,…A_nn are square and all its blocks except A₁₂A₂₃,…A_h-1,hA_h1 are zero. It is shown that a nonnegative matrix in this form is irreducible with index of imprimitivity hif and only if it has no zero rows nor columns and the product A₁₂A₂₃… A_h1 is primitive. It is also shown that if Ais nonsingular, then each of the blocks A₁₂, A₂₃,…, A_h1 is n/h-square.