On Perron complements of totally nonnegative matrices

An n x n matrix is called totally nonnegative if every minor of A is nonneg ative, The problem of interest is to describe the Perron complement of a pr incipal submatrix of an irreducible totally nonnegative matrix. We show tha t the Perron complement of a totally nonnegative matrix is totally nonnegat ive only if the complementary index set is based on consecutive indices. We also demonstrate a quotient formula for Perron complements analogous to th e so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement of totally nonnegative m atrices, when the Perron complement is totally nonnegative. (C) 2001 Elsevi er Science Inc. All rights reserved.