Differential Topology and Matrix Analysis: An Overview

This overview illustrates the application of methods from differential topology to several 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 the 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-posedness and its relation to problem conditioning. The conditioning of matrix problems is considered in both deterministic and probabilistic settings. The paper also addresses the regularization of ill-posed matrix problems in the presence of errors. Several examples are provided to illustrate these concepts and their practical relevance. The overview is intended for specialists from different fields who use matrix analysis in their work and do not have a strong background in differential topology.