A Class of Robustness Problems in Matrix Analysis
André C. M. Ran, Leiba Rodman · Birkhäuser Basel eBooks · 2002
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 X ∈ A , a set G ( X ) is given. An element Y 0 ∈ G ( X 0 ) will be called robust (relative to the sets A and G ( X ) if for every X ∈ A close enough to X 0 there is a X ∈ G ( 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 0 ‖ 1/α . 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.