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 Cᵏ 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 Cᵏ⁻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 (θ) 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.