Simultaneous solutions of matrix equations and simultaneous equivalence of matrices

Given matrices Ai, Bi and Ci(i∈I) of corresponding dimensions over a field F, we prove that: (i) if are simultaneously similar to , then there exists a simultaneous solution X to the matrix Sylvester equations AiX-XBi=Ci; and (ii) if are simultaneously equivalent to then there exist simultaneous solutions X,Y to the matrix equations AiX-YBi=Ci.We also show that analogous results hold for mixed pairs of matrix Sylvester equations A1X1-YB1=C1, A2X2-YB2=C2 and for generalized Stein equations X-AYB=C.