Copyright © 2013 Shao-Wen Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression where is a Hermitian solution to quaternion matrix equations , , and . As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations , , , and , which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement with respect to a Hermitian g-inverse of , which is a common solution to quaternion matrix equations and , are also considered.