A new solvable condition for a pair of generalized Sylvester equations

Abstract

A necessary and sufficient condition is given for the quaternion matrix equations
A_iX + Y B_i = C_i (i = 1, 2) to have a pair of common solutions X and Y . As a consequence, the
results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear
matrix equation AX + Y B = C, Comput. Math. Appl., 52 (2006), pp. 861-872).