Critical Points of the Singular Value Decomposition

The singular value decomposition (SVD) is a factorization that is discontinuous on the subset of matrices having repeated singular values. In this paper the SVD is studied in the vicinity of this critical set. Each one-parameter $C^k$ perturbation transversal to the critical set is shown to uniquely determine an SVD at the critical point that extends to an SVD along the perturbation path that is $C^{k-1}$ in the perturbation parameter. Derivatives of the singular vectors at the critical point are found explicitly. Application is made to the effect on the singular vectors of perturbations from a matrix in the critical set and compared to the information provided by the $\sin (\theta)$ theorem. Estimates of the derivative of the singular vectors are applied to inequalities involving the matrix absolute value, such as the generalized Araki--Yamagami inequality.