Stability in Matrix Analysis Problems

Let there be given a class of matrices $$\mathcal{A}$$ , and for each $$X \in \mathcal{A}$$ , a set $$\mathcal{G}(X)$$ . The following two broadly formulated problems are addressed: 1. An element $$Y _{0} \in \mathcal{G}(X_{0})$$ will be called stable with respect to $$\mathcal{A}$$ and the collection $$\{\mathcal{G}(X)\}_{X\in \mathcal{A}}$$ , if for every $$X \in \mathcal{A}$$ which is sufficiently close to X 0 there exists an element $$Y \in \mathcal{G}(X)$$ that is as close to Y 0 as we wish. Give criteria for existence of a stable Y 0, and describe all of them. 2. Fix α ≥ 1. An element $$Y _{0} \in \mathcal{G}(X_{0})$$ will be called α-stable with respect to $$\mathcal{A}$$ and the collection $$\{\mathcal{G}(X)\}_{X\in \mathcal{A}}$$ , if for every $$X \in \mathcal{A}$$ which is sufficiently close to X 0 there exists an element $$Y \in \mathcal{G}(X)$$ such that the distance between Y and Y 0 is bounded above by a constant times the distance between X and X 0 raised to the power 1∕α where the constant does not depend on X. Give criteria for existence of an α-stable Y 0, and describe all of them. We present an overview of several basic results and literature guide concerning various stability notions, including the concept of conditional stability, as well as prove several new results and state open problems of interest. Large part of the work leading to this chapter was done while the second author visited Vrije Universiteit, Amsterdam, whose hospitality is gratefully acknowledged.