Entrywise preservers of sign regularity

Projesh Nath Choudhury; Shivangi Yadav

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arXiv · 2509.17902

Entrywise preservers of sign regularity

Projesh Nath Choudhury·Shivangi Yadav

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Abstract

Entrywise functions preserving positivity and related notions have a rich history, beginning with the seminal works of Schur, Pólya-Szegő, Schoenberg, and Rudin. Following their classical results, it is well-known that entrywise functions preserving positive semidefiniteness for matrices of all dimensions must be real analytic with non-negative Taylor coefficients. These works were taken forward in the last decade by Belton, Guillot, Khare, Putinar, and Rajaratnam. Recently, Belton-Guillot-Khare-Putinar [J. d'Analyse Math. 2023] classified all functions that entrywise preserve totally positive (TP) and totally non-negative (TN) matrices. In this paper, we study entrywise preservers of strictly sign regular and sign regular matrices - a class that includes TP/TN matrices as special cases and was first studied by Schoenberg in 1930 to characterize variation diminution. Our main results provide complete characterizations of entrywise transforms of rectangular matrices which preserve: (i)~sign regularity and strict sign regularity, as well as (ii)~sign regularity and strict sign regularity with a given sign pattern.

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Projesh Nath Choudhury, Shivangi Yadav. 2026-04-24. Entrywise preservers of sign regularity. https://arxiv.org/abs/2509.17902

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