On totally nonpositive matrices associated with a triple negatively realizable

Abstract

Let $A \in {\mathbb {R}}^{n \times n}$ be a totally nonpositive matrix (t.n.p.) with rank r and principal rank p, that is, every minor of A is nonpositive and p is the size of the largest invertible principal submatrix of A. We introduce that a triple (n, r, p) will be called negatively realizable if there exists a t.n.p. matrix A of order n and such that its rank is r and its principal rank is p. In this work we extend the results obtained for irreducible totally nonnegative matrices given in Cantó and Urbano (Linear Algebra Appl 551:125–146. https://doi.org/10.1016/j.laa.2018.03.045, 2018) to t.n.p. matrices. For that, we consider the sequence of the first p-indices of A and study the linear dependence relations between their rows and columns. These relations allow us to construct t.n.p. matrices associated with a triple (n, r, p) negatively realizable and a specific sequence of the first p-indices.

Appendix

In this section, we construct Algorithm 2 associated with Procedure 1 to obtain an upper block echelon TN matrix $V\in {\mathbb {R}}^{n \times n}$ with $\mathop {\mathrm{rank}}\nolimits (V)=r$, $p$-$\mathop {\mathrm{rank}}\nolimits (V)=p$ and the sequence of its first p-indices given by $\{ 1,i_2,\ldots ,i_p \}$. To apply Algorithm 2 we introduce Algorithm 1 given in [4, Algorithm 1] in order to know the maximum rank of a matrix depending on the sequence of its first p-indices.

Finally, we consider a triple $(n,r,p) (1,i_2,\ldots ,i_p)$-negatively realizable of the type-I (type-II), and we construct Algorithm 3 (Algorithm 4) to obtain a type-I (type-II) t.n.p. matrix A associated with the given triple. These algorithms are based on Procedure 2.

As we have seen in the proof of Proposition 8 to obtain a t.n.p. matrix by Procedure 2 we only need that $\, d_{1} \ge \sum _{j=2}^{i_p} v_{jn}$. As a consequence, from Algorithms 3 and 4, we can obtain different type-I and type-II t.n.p. matrices associated with the given triple by changing the value of D(1, 1) and whenever $D(1,1) \le -t*V(:,n)$.