Jordan structures of irreducible totally nonnegative matrices with a prescribed sequence of the first p-indices

Abstract

Let A in {mathbb {R}}^{n times n} be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, A is irreducible with all minors nonnegative, r is the size of the largest invertible square submatrix of A and p is the size of its largest invertible principal submatrix. We consider the sequence {1,i_2,ldots ,i_p} of the first p-indices of A as the first initial row and column indices of a p times p invertible principal submatrix of A. A triple (n, r, p) is called (1,i_2,ldots ,i_p)-realizable if there exists an irreducible totally nonnegative matrix A in {mathbb {R}}^{n times n} with rank r, principal rank p, and {1,i_2,ldots ,i_p} is the sequence of its first p-indices. In this work we study the Jordan structures corresponding to the zero eigenvalue of irreducible totally nonnegative matrices associated with a triple (n, r, p) (1,i_2,ldots ,i_p)-realizable.