A Class of Robustness Problems in Matrix Analysis

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 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.