Abstract:
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.