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

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.