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Zbl 1223.90077
Liu, Yonghui; Tian, Yongge
Max-min problems on the ranks and inertias of the matrix expressions $A - BXC \pm (BXC)^{\ast}$ with applications. (English)
[J] J. Optim. Theory Appl. 148, No. 3, 593-622 (2011). ISSN 0022-3239; ISSN 1573-2878/e

In this paper, a simultaneous decomposition for a matrix triplet $(A,B,C ^{\ast})$ is introduced, where $A=\pm A^{\ast}$ and $(\cdot )^{\ast}$ denotes the conjugate transpose of a matrix. Some conjectures on the maximal and minimal values of the ranks of the matrix expressions $A - BXC \pm (BXC)^{\ast}$ with respect to a variable matrix $X$ are solved. Some explicit formulas for the maximal and minimal values of the inertia of the matrix expression $A - BXC - (BXC)^{\ast}$ with respect to $X$ are given. As applications, the extremal ranks and inertias of the matrix expression $D - CXC^{\ast}$ subject to Hermitian solutions of a consistent matrix equation $AXA^{\ast}=B$, as well as the extremal ranks and inertias of the Hermitian Schur complement $D - B^{\ast} A ^{\sim} B$ with respect to a Hermitian generalized inverse $A ^{\sim}$ of $A$ are derived.
[Zhang Xian (Xiamen)]

MSC 2000:: *90C47 Minimax problems
15A09 Matrix inversion

Keywords: Hermitian matrix; rank; inertia; generalized inverse; Schur complement; inequality

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Zentralblatt MATH Berlin [Germany]