Sign regularity preserving linear operators

A matrix A?Rm�n is strictly sign regular/SSR (or sign regular/SR) if for each 1?k?min{m,n}, all k�k minors of A (or non-zero k�k minors of A) have the same sign. This class of matrices contains the totally positive matrices, and was first studied by Schoenberg (1930) to characterize Variation Diminution (VD), a fundamental property in total positivity theory. In this note, we classify all surjective linear mappings L:Rm�n?Rm�n that preserve: (i) sign regularity and (ii) sign regularity with a given sign pattern, as well as (iii) strict versions of these.