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

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Variation diminution (VD) is a fundamental property in total positivity theory, first studied in 1912 by Fekete–Pólya for one-sided Pólya 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 × n m \times n matrices for m > n m>n using this property. In this article we strengthen these results by characterizing all m × n 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–Pólya [Rend. Circ. Mat. Palermo 34 (1912), pp. 89–120] and continued by Schoenberg [Math. Z. 32 (1930), pp. 321–328], Motzkin [Beiträge zur Theorie der linearen Ungleichungen, PhD Dissertation, Jerusalem, 1936], Gantmacher–Krein [Oscillyacionye matricy i yadra i malye kolebaniya mehaničeskih sistem, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950], Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 76 (1981), pp. 824–832], and Choudhury [Bull. Lond. Math. Soc. 54 (2022), pp. 791–811; Bull. Sci. Math. 186 (2023), p. 21]. 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.