Journal of Applied Mathematics
Volume 2013 (2013), Article ID 514984, 9 pages
Research Article

Ranks of a Constrained Hermitian Matrix Expression with Applications

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

Received 12 November 2012; Accepted 4 January 2013

Academic Editor: Yang Zhang

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.


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.