The maximal and minimal ranks of a quaternion matrix expression with applications

In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X1, X2) = C7 − A4X1B4 − A5X2B5 where X1, X2 are variant quaternion matrices subject to quaternion matrix equations A1X1 = C1, A2X1 = C2, A3X1 = C3, X2B1 = C4, X2B2 = C5, X2B3 = C6. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper.