in this paper, we study the extremal ranks and inertias of the hermitian matrix expression $$ f(x,y)=c_{4}-b_{4}y-(b_{4}y)^{*}-a_{4}xa_{4}^{*},$$ where $c_{4}$ is hermitian, $*$ denotes the conjugate transpose, $x$ and $y$ satisfy the following consistent system of matrix equations $a_{3}y=c_{3}, a_{1}x=c_{1},xb_{1}=d_{1},a_{2}xa_{2}^{*}=c_{2},x=x^{*}.$ as consequences, we g...