Sign regularity preserving linear operators

A matrix (Formula presented.) is strictly sign regular (or sign regular) if for each (Formula presented.), all (nonzero) (Formula presented.) minors of (Formula presented.) have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg in 1930 to characterize variation diminution, a fundamental property in total positivity theory. In this article, we classify all surjective linear mappings (Formula presented.) that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.