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.Comment: 16 page