Sign regularity preserving linear operators
Projesh Nath Choudhury, Shivangi Yadav
TL;DR
The paper addresses the problem of classifying surjective linear maps on the space $\mathbb{R}^{m\times n}$ that preserve sign-regular (SR) and strictly sign-regular (SSR) matrices, including fixed-sign-pattern variants $SR(\epsilon)$ and $SSR(\epsilon)$ across all sizes. It shows that preservers are precisely compositions of simple, structure-preserving operations: $A\mapsto FAE$ with $F,E$ diagonal and positive, $A\mapsto -A$, $A\mapsto P_mA$, $A\mapsto AP_n$, and, when $m=n$, $A\mapsto A^T$; crucially, preserving $SR$ reduces to preserving $SR_2$. The results extend the classical TP/TN preservers (BHJ85) to arbitrary sizes and patterns, and include a complete treatment of the $2\times2$ case and the fixed-sign-pattern variants, supported by density arguments and linear-algebraic structure. Altogether, the work advances linear preserver theory for positivity classes and connects to variation-diminishing properties in total positivity theory.
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.