Investigation on the Hermitian matrix expression‎ ‎subject to some consistent equations

Abstract

In this paper‎, ‎we study the extremal‎ <br />‎ranks and inertias of the Hermitian matrix expression $$‎ <br />‎f(X,Y)=C_{4}-B_{4}Y-(B_{4}Y)^{*}-A_{4}XA_{4}^{*},$$ where $C_{4}$ is‎ <br />‎Hermitian‎, ‎$*$ denotes the conjugate transpose‎, ‎$X$ and $Y$ satisfy‎ <br />‎the following consistent system of matrix equations $A_{3}Y=C_{3}‎, <br />‎A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2},X=X^{*}.$ As‎ <br />‎consequences‎, ‎we get the necessary and sufficient conditions for the‎ <br />‎above expression $f(X,Y)$ to be (semi) positive‎, ‎(semi) negative‎. <br />‎The relations between the Hermitian part of the solution to the‎ <br />‎matrix equation $A_{3}Y=C_{3}$ and the Hermitian solution to the‎ <br />‎system of matrix equations‎ <br />‎$A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2}$ are also‎ <br />‎characterized‎. ‎Moreover‎, ‎we give the necessary and sufficient‎ <br />‎conditions for the solvability to the‎ <br />‎following system of matrix equations‎ <br />‎$A_{3}Y=C_{3},A_{1}X=C_{1},XB_{1}=D_{1}‎, <br />‎A_{2}XA_{2}^{*}=C_{2},X=X^{*}‎, <br />‎B_{4}Y+(B_{4}Y)^{*}+A_{4}XA_{4}^{*}=C_{4} $ and provide an‎ <br />‎expression of the general solution to this system‎ <br />‎when it is solvable‎.