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−A4XA4∗ where X is a Hermitian solution to quaternion matrix equations A1X=C1, XB1=C2, and A3XA3*=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, A3XA3*=C3, and A4XA4*=C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement C4−A4A3~A4∗ with respect to a Hermitian g-inverse A3~ of A3, which is a common solution to quaternion matrix equations A1X=C1 and XB1=C2, are also considered