A generalization of doubly stochastic matrices over a field

Abstract

Let X and Y be m× n matrices over a field F such that Y T X is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and Y TA = K ̂ Y T where K ϵ Λ and K ̂ ϵ Λ′ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ( Λ, Λ′) of all such solutions.