Ranks of Solutions of the Linear Matrix Equation AX + YB = C

For a consistent complex matrix equation AX + YB = C, we solve the following two problems: (1) the maximal and minimal ranks of a pair of solutions X and Y to AX + YB = C, and (2) the maximal and minimal ranks of four real matrices X0, X1, Y0, and Y1 in a pair of solutions X = X0 + iX1 and Y = Y0 + iY1 to AX + YB = C. We also give a necessary and sufficient condition for matrix equations AiXi + YiBi = C (i = 1, 2) to have common solutions.