Constructing strictly sign regular matrices of all sizes and sign patterns

Abstract

The class of strictly sign regular (SSR) matrices has beenextensively studied by many authors over the past cen-tury, notably by Schoenberg, Motzkin, Gantmacher, andKrein. A classical result of Gantmacher–Krein assuresthe existence of SSR matrices for any dimension andsign pattern. In this article, we provide an algorithm toexplicitly construct an SSR matrix of any given size andsign pattern. (We also provide in the Appendix, a Pythoncode implementing our algorithm.) To develop this algo-rithm, we show that one can extend an SSR matrix byadding an extra row (column) to its border, resultingin a higher order SSR matrix. Furthermore, we showhow inserting a suitable new row/column between anytwo successive rows/columns of an SSR matrix resultsin a matrix that remains SSR. We also establish analo-gous results for SSR 𝑚 × 𝑛 matrices of order 𝑝 for any𝑝 ∈ [1, min{𝑚, 𝑛}].