Differential Topology and Matrix Analysis: An Overview

The present overview aims to illustrate the application of differential topology methods to some important problems in matrix analysis. In particular, it focuses on the use of smooth manifolds and smooth mappings to study fundamental issues such as the determination of matrix rank and the computation of the Jordan form in presence of uncertainties. Various aspects of numerical matrix analysis are discussed, including the genericity of matrix problems, characterization of singular sets in the parameter space, the distance to ill-posed problems and its relation to problem conditioning. The paper also addresses the conditioning of matrix problems in both deterministic and probabilistic settings and the regularization of ill--posed matrix problems. Several examples are provided to illustrate these concepts and their practical relevance.