A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ 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.