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

Abstract

AbstractLet $$A \in {\mathbb {R}}^{n \times n}$$ A ∈ R n × 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 $$\{1,i_2,\ldots ,i_p\}$$ { 1 , i 2 , … , i p } of the first p-indices of A as the first initial row and column indices of a $$p \times p$$ p × p invertible principal submatrix of A. A triple (n, r, p) is called $$(1,i_2,\ldots ,i_p)$$ ( 1 , i 2 , … , i p ) -realizable if there exists an irreducible totally nonnegative matrix $$A \in {\mathbb {R}}^{n \times n}$$ A ∈ R n × n with rank r, principal rank p, and $$\{1,i_2,\ldots ,i_p\}$$ { 1 , i 2 , … , 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) $$(1,i_2,\ldots ,i_p)$$ ( 1 , i 2 , … , i p ) -realizable.