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.

翻译：本文重新探讨了[J.-G. Sun, Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math. 73(2), pp. 235--263, 1996]一文中关于计算左右奇异子空间的条件数。对于实矩阵与复矩阵，我提出了一种替代性的计算方法，该计算在矩阵输入空间采用欧几里得距离，在输出线性子空间Grassmann流形上采用弦距离、Grassmann距离及Procrustes距离。与精确值相比，该条件数至多相差一个小的因子，其等于选定奇异子空间对应奇异值与未选定奇异值之间的最小奇异值间隔的倒数。