Interval hulls of $N$-matrices and almost $P$-matrices

TL;DR

This paper characterizes almost P-matrices and N-matrices using a sign non-reversal property, providing a finite test to determine if all matrices within an interval hull possess these properties, extending to semipositive matrices.

Contribution

It introduces a finite criterion based on sign non-reversal to verify if all matrices in an interval hull are N-matrices or almost P-matrices, generalizing previous matrix class characterizations.

Findings

Finite subset of interval hull determines matrix class membership

Characterizations extend to semipositive and minimally semipositive matrices

Provides a practical test for entire classes of matrices within an interval

Abstract

We establish a characterization of almost $P$-matrices via a sign non-reversal property. In this we are inspired by the analogous results for $N$-matrices. Next, the interval hull of two $m \times n$ matrices $A=(a_{ij})$ and $B = (b_{ij})$, denoted by $\mathbb{I}(A,B)$, is the collection of all matrices $C \in \mathbb{R}^{m \times n}$ such that each $c_{ij}$ is a convex combination of $a_{ij}$ and $b_{ij}$. Using the sign non-reversal property, we identify a finite subset of $\mathbb{I}(A,B)$ that determines if all matrices in $\mathbb{I}(A,B)$ are $N$-matrices/almost $P$-matrices. This provides a test for an entire class of matrices simultaneously to be $N$-matrices/almost $P$-matrices. We also establish analogous results for semipositive and minimally semipositive matrices. These characterizations may be considered similar in spirit to that of $P$-matrices by Bialas-Garloff [Linear Algebra Appl. 1984] and Rohn-Rex [SIMAX 1996], and of positive definite matrices by Rohn [SIMAX 1994].