The condition number of singular subspaces, revisited

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.