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A Class of Robustness Problems in Matrix Analysis

  • André C. M. Ran
  • Leiba Rodman
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

Abstract

We present an overview of several results and a literature guide, prove some new results, and state open problems concerning description of all robust matrices in the following sense: Let be given a class of real or complex matrices A, and for each XA, a set G(X) is given. An element Y 0G(X 0) will be called robust (relative to the sets A and G(X) if for every XA close enough to X 0 there is a XG(X) that is as close to Y 0 as we wish. The following topics are covered, with respect to the robustness property: 1. Invariant subspaces of matrices; here the set G(X) is the set of all X-invariant subspaces. 2. Invariant subspaces of matrices with symmetries related to indefinite inner products. The invariant subspaces in question include semidefinite and neutral subspaces (with respect to an indefinite inner product). 3. Applications of invariant subspaces of matrices with or without symmetries. The applications include: general matrix quadratic equations, the continuous and discrete algebraic Riccati equations, minimal factorization of rational matrix functions with symmetries and the transport equation from mathematical physics. 4. Several matrix decompositions: polar decompositions with respect to an indefinite inner product, Cholesky factorizations, singular value decomposition.

Other related notions of robustness are studied as well, for example, a stronger notion of α-robustness, in which the magnitude of degree of closeness of Y and Y 0 (as measured in some appropriate metric) does not exceed the magnitude of ‖X-X 01/α.

Keywords

Invariant Subspace Polar Decomposition Cholesky Decomposition Algebraic Riccati Equation Lagrangian Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • André C. M. Ran
    • 1
  • Leiba Rodman
    • 2
  1. 1.Divisie Wiskunde en Informatica Faculteit Exacte WetenschappenVrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Department of MathematicsThe College of William and MaryWilliamsburgUSA

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