Investigation on the Hermitian matrix expression‎ ‎subject to some consistent equations


Department of Mathematics, Zunyi Normal College Shanghai Road, Zunyi 563000, P.R. China


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
‎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.


Main Subjects