Sign regular matrices and variation diminution: single-vector tests and characterizations, following Schoenberg, Gantmacher-Krein, and Motzkin

Abstract: Variation diminution (VD) is a fundamental property in total positivity theory, first studied in 1912 by Fekete-P\'olya for one-sided P\'olya frequency sequences, followed by Schoenberg, and by Motzkin who characterized sign regular (SR) matrices using VD and some rank hypotheses. A classical theorem by Gantmacher-Krein characterized the strictly sign regular (SSR) $m \times n$ matrices for $m>n$ using this property. In this article we strengthen these results by characterizing all $m \times n$ SSR matrices using VD. We further characterize strict sign regularity of a given sign pattern in terms of VD together with a natural condition motivated by total positivity. We then refine Motzkin's characterization of SR matrices by omitting the rank condition and specifying the sign pattern. This concludes a line of investigation on VD started by Fekete-P\'olya [Rend. Circ. Mat. Palermo 1912] and continued by Schoenberg [Math. Z. 1930], Motzkin [PhD thesis, 1936], Gantmacher-Krein [1950 book], Brown-Johnstone-MacGibbon [J. Amer. Stat. Assoc. 1981], and Choudhury [Bull. London Math. Soc. 2022, Bull. Sci. Math. 2023]. In fact we show stronger characterizations, by employing single test vectors with alternating sign coordinates - i.e., lying in the alternating bi-orthant. We also show that test vectors chosen from any other orthant will not work.