  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.
