Ranks of a Constrained Hermitian Matrix Expression with Applications.

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression C4 - A4XA*4 where X is a Hermitian solution to quaternion matrix equations A1X = C1, XB1 = C2, and A3XA*3 = C3. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations A1X = C1, XB1 =C2, A3XA*3 = C3, and A4XA*4 = C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement C4 - A4Ã3A*4 with respect to a Hermitian g-inverse Ã3 of A3, which is a common solution to quaternion matrix equations A1X = C1 and XB1= C2, are also considered. [ABSTRACT FROM AUTHOR]