Max-min problems on the ranks and inertias of the matrix expressions \(A - BXC \pm (BXC)^{\ast}\) with applications

DOI10.1007/s10957-010-9760-8zbMath1223.90077OpenAlexW2148085383MaRDI QIDQ535072

Yongge Tian, Yong Hui Liu

Publication date: 11 May 2011

Published in: Journal of Optimization Theory and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10957-010-9760-8

zbMATH Keywords

Schur complementinequalityrankgeneralized inverseinertiaHermitian matrix

Mathematics Subject Classification ID

Minimax problems in mathematical programming (90C47) Theory of matrix inversion and generalized inverses (15A09)

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Cites Work

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• A complete algebraic solvability test for the nonstrict Lyapunov inequality
• Positive definite completions of partial Hermitian matrices
• The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample
• The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems
• Equalities and inequalities for inertias of Hermitian matrices with applications
• Ranks of Hermitian and skew-Hermitian solutions to the matrix equation \(AXA^*=B\)
• Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic
• Restricted quadratic forms, inertia theorems, and the Schur complement
• Symmetrizers of matrices
• The inertia of a Hermitian matrix having prescribed complementary principal submatrices
• The inertia of a Hermitian matrix having prescribed diagonal blocks
• One-step completions of Hermitian partial matrices with minimal negative signature
• The negative signature of some Hermitian matrices
• Maximum rank matrix completion
• Toeplitz minimal rank completions
• State covariance assignment problem with measurement noise: A unified approach based on a symmetric matrix equation
• All controllers for the general \({\mathcal H}_ \infty\) control problem: LMI existence conditions and state space formulas
• The inertia of certain Hermitian block matrices
• The maximal and minimal ranks of \(A - BXC\) with applications
• Completing triangular block matrices with maximal and minimal ranks
• The possible inertias for a Hermitian matrix and its principal submatrices
• Minimal rank completions for block matrices
• The maximal and minimal ranks of some expressions of generalized inverses of matrices
• Upper and lower bounds for ranks of matrix expressions using generalized inverses
• Generalized inverses. Theory and applications.
• Invertible completions of operator matrices
• Partially specified matrices and operators: classification, completion, applications
• Extremal ranks of submatrices in an Hermitian solution to the matrix equation \(AXA^{*}=B\) with applications
• Quadratic inequalities for Hilbert space operators
• Rank equalities for idempotent matrices with applications.
• Determination of the inertia of a partitioned Hermitian matrix
• On the interia of some classes of partitioned matrices
• More on maximal and minimal ranks of Schur complements with applications
• Rank equalities related to outer inverses of matrices and applications
• A simultaneous decomposition of a matrix triplet with applications
• Completing block Hermitian matrices with maximal and minimal ranks and inertias
• On the Complexity of Matrix Rank and Rigidity
• The complexity of matrix completion
• More on extremal ranks of the matrix expressions A  − BX  ± X ^* B ^* with statistical applications
• Matrix Analysis
• Towards a Generalized Singular Value Decomposition
• Assigning controllability and observability Gramians in feedback control
• The inertia of hermitian matrices with a prescribed 2×2 block decomposition
• Minimal rank completions of partial banded matrices
• A linear matrix inequality approach to H_∞ control
• Hermitian and normal completions
• Inertias of Block Band Matrix Completions
• Ranks of Solutions of the Matrix Equation AXB = C
• The Minimum Rank of a 3 × 3 Partial Block Matrix
• Extremal Ranks of Some Symmetric Matrix Expressions with Applications
• Rank equalities for idempotent and involutory matrices