When does the inverse have the same sign pattern as the transpose?

Abstract

By a sign pattern (matrix) we mean an array whose entries are from the set {+, −, 0}. The sign patterns A for which every real matrix with sign pattern A has the property that its inverse has sign pattern A T are characterized. Sign patterns A for which some real matrix with sign pattern A has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices is examined.