Sign regularity preserving linear operators

TL;DR

This paper classifies all surjective linear operators on matrices that preserve various notions of sign regularity, including strict and pattern-specific cases, extending the understanding of total positivity and variation diminution.

Contribution

It provides a complete classification of linear maps preserving sign regularity and its variants, a significant extension in matrix analysis and total positivity theory.

Findings

Characterization of linear preservers of sign regularity

Extension to strict and pattern-specific sign regularity

Implications for total positivity and variation diminution

Abstract

A matrix $A\in \mathbb{R}^{m \times n}$ is strictly sign regular/SSR (or sign regular/SR) if for each $1 \leq k \leq \min\{m,n\}$, all (non-zero) $k\times k$ minors of $A$ have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg in 1930 to characterize variation diminution, a fundamental property in total positivity theory. In this article, we classify all surjective linear mappings $\mathcal{L}:\mathbb{R}^{m\times n}\to\mathbb{R}^{m\times n}$ that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.