On the existence of a common solution X to the matrix equations AiXBj=Cij, (i,j) Γ

In this paper conditions are derived for the existence of a common solution X to the matrix equations AiXBj=Cij,(i,j) Γ, where the matrices Ai,Bj,Cij and X have suitable dimensions and the (i,j)’s are index pairs in some set Γ. The purpose of this paper is to present, for certain specific sets of index pairs Γ, verifiable necessary and sufficient solvability conditions that are stated directly in terms of the matrices and that do not use Kronecker products.