Abstract :
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 get the necessary and sufficient conditions for the
above expression $f(X,Y)$ to be (semi) positive, (semi) negative.
The relations between the Hermitian part of the solution to the
matrix equation $A_{3}Y=C_{3}$ and the Hermitian solution to the
system of matrix equations
$A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2}$ are also
characterized. Moreover, we give the necessary and sufficient
conditions for the solvability to the
following 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^{*},
B_{4}Y+(B_{4}Y)^{*}+A_{4}XA_{4}^{*}=C_{4} $ and provide an
expression of the general solution to this system
when it is solvable.
