Takagi factorization of matrices depending on parameters and locating degeneracies of singular values

TL;DR

This paper studies the Takagi factorization of parameter-dependent matrices, analyzing the smoothness, genericity, and degeneracies of singular values, and develops numerical methods to locate and study these degeneracies.

Contribution

It provides theoretical results on the co-dimension of singular value degeneracies and introduces numerical techniques to identify these phenomena in parameter space.

Findings

Degeneracies have co-dimension 2.

Numerical methods effectively locate degeneracies.

Density of degeneracies analyzed numerically.

Abstract

In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to $0$ or multiple singular values. For these phenomena, we give theoretical results showing that their co-dimension is $2$, and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.