Ranks of a Constrained Hermitian Matrix Expression with Applications

Abstract

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression $C_4-A_{4}X{A}_4^{\ast }$ where $X$ is a Hermitian solution to quaternion matrix equations $A_{1}X=C_1$, $X{B}_1=C_2$, and $A_{3}X{A}_3^{*}=C_3$. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations $A_{1}X=C_1$, $X{B}_1=C_2$, $A_{3}X{A}_3^{*}=C_3$, and $A_{4}X{A}_4^{*}=C_4$, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement $C_4-A_4{A}_3^{~}{A}_4^{\ast }$ with respect to a Hermitian g-inverse $A_3^{~}$ of $A_3$, which is a common solution to quaternion matrix equations $A_{1}X=C_1$ and $X{B}_1=C_2$, are also considered.

Erratum to “Ranks of a Constrained Hermitian Matrix Expression with Applications” dx.doi.org/10.1155/2013/826397