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

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.

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