The condition number of singular subspaces, revisited

TL;DR

This paper revisits the condition number for computing singular subspaces, providing an alternative calculation method that relates to singular value gaps and various geometric distances on Grassmann manifolds.

Contribution

It introduces a new way to compute the condition number for singular subspaces using Euclidean and Grassmannian distances, linking it to singular value gaps.

Findings

Condition number equals inverse singular value gap up to a small factor.

Provides alternative computation methods for real and complex matrices.

Connects condition number with geometric distances on Grassmann manifolds.

Abstract

I revisit the condition number of computing left and right singular subspaces from [J.-G. Sun, Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math. 73(2), pp. 235--263, 1996]. For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values corresponding to the selected singular subspace and those not selected.