Constructing strictly sign regular matrices of all sizes and sign patterns

TL;DR

This paper presents an explicit algorithm for constructing strictly sign regular matrices of any size and sign pattern, extending existing theoretical results with practical methods and implementation code.

Contribution

It introduces a constructive algorithm for SSR matrices of arbitrary size and sign pattern, including methods to extend and modify SSR matrices while preserving their properties.

Findings

Algorithm successfully constructs SSR matrices for all sizes and patterns.

Adding rows or columns can extend SSR matrices while maintaining their properties.

Provides Python implementation of the construction algorithm.

Abstract

The class of strictly sign regular (SSR) matrices has been extensively studied by many authors over the past century, notably by Schoenberg, Motzkin, Gantmacher, and Krein. A classical result of Gantmacher-Krein assures the existence of SSR matrices for any dimension and sign pattern. In this article, we provide an algorithm to explicitly construct an SSR matrix of any given size and sign pattern. (We also provide in an Appendix, a Python code implementing our algorithm.) To develop this algorithm, we show that one can extend an SSR matrix by adding an extra row (column) to its border, resulting in a higher order SSR matrix. Furthermore, we show how inserting a suitable new row/column between any two successive rows/columns of an SSR matrix results in a matrix that remains SSR. We also establish analogous results for strictly sign regular $m \times n$ matrices of order $p$ for any $p \in [1, \min\{m,n\}]$.