On Perron complements of totally nonnegative matrices

AbstractAn n×n matrix is called totally nonnegative if every minor of A is nonnegative. The problem of interest is to describe the Perron complement of a principal submatrix of an irreducible totally nonnegative matrix. We show that the Perron complement of a totally nonnegative matrix is totally nonnegative only if the complementary index set is based on consecutive indices. We also demonstrate a quotient formula for Perron complements analogous to the so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement of totally nonnegative matrices, when the Perron complement is totally nonnegative