Department of Mathematics, Zunyi Normal College Shanghai Road, Zunyi 563000, P.R. China
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.
Zhang, X. (2014). Investigation on the Hermitian matrix expression subject to some consistent equations. Bulletin of the Iranian Mathematical Society, 40(1), 9-28.
MLA
Xiang Zhang. "Investigation on the Hermitian matrix expression subject to some consistent equations". Bulletin of the Iranian Mathematical Society, 40, 1, 2014, 9-28.
HARVARD
Zhang, X. (2014). 'Investigation on the Hermitian matrix expression subject to some consistent equations', Bulletin of the Iranian Mathematical Society, 40(1), pp. 9-28.
VANCOUVER
Zhang, X. Investigation on the Hermitian matrix expression subject to some consistent equations. Bulletin of the Iranian Mathematical Society, 2014; 40(1): 9-28.