RANKS OF THE COMMON SOLUTION TO SOME QUATERNION MATRIX EQUATIONS WITH APPLICATIONS

We derive the formulas of the maximal and minimal ranks of four real matrices X1, X2, X3 and X4 in common solution X = X1 + X2i + X3j + X4k to quaternion matrix equations A1 X = C1, X B2 = C2, A3 X B3 = C3. As applications, we establish necessary and sufficient conditions for the existence of the common real and complex solutions to the matrix equations. We give the expressions of such solutions to this system when the solvability conditions are met. Moreover, we present necessary and sufficient conditions for the existence of real and complex solutions to the system of quaternion matrix equations A1 X = C1, X B2 = C2, A3 X B3 = C3, A4 X B4 = C4. The findings of this paper extend some known results in the literature.