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Variation diminution (VD) is a fundamental property in total positivitytheory, first studied by Fekete-P\'olya (1912) for one sided P\'olya frequencysequences, followed by Schoenberg (1930), and by Motzkin (1936) whocharacterized sign regular (SR) matrices using VD and some rank hypotheses. Aclassical theorem in 1950 by Gantmacher-Krein characterized the strictly signregular (SSR) $m \times n$ matrices for $m>n$ using this property. In thisarticle we strengthen their result by characterizing all $m \times n$ SSRmatrices using VD. We further characterize strict sign regularity of a givensign pattern in terms of VD together with a natural condition motivated bytotal positivity. We then refine Motzkin's characterization of SR matrices byomitting the rank condition and specifying the sign pattern. More strongly,these characterizations employ single test vectors with alternating signcoordinates - i.e., lying in an alternating bi-orthant.  The second contribution of our work includes the study of linear preserversof SR and SSR matrices. The linear preserver problem is an important questionin matrix theory and operator theory. We classify all 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 wellas strict versions of these.

Sign regularity: Refinement of Gantmacher-Krein-Motzkin results on variation diminution and the linear preserver problem