Max-Min Problems on the Ranks and Inertias of the Matrix Expressions ABXC±(BXC) with Applications



We introduce a simultaneous decomposition for a matrix triplet (A,B,C ), where AA and (⋅) denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions ABXC±(BXC) with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression ABXC−(BXC) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression DCXC subject to Hermitian solutions of a consistent matrix equation AXA =B, as well as the extremal ranks and inertias of the Hermitian Schur complement DB A B with respect to a Hermitian generalized inverse A of A. Various consequences of these extremal ranks and inertias are also presented in the paper.


Hermitian matrix Rank Inertia Generalized inverse Schur complement Inequality 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsShanghai Finance UniversityShanghaiChina
  2. 2.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina

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