Conditioning

The singular values and singular vectors of $A$ have largely the same conditioning properties as the eigenvalues and eigenvectors of a Hermitian matrix, as described in §2.2.5.

The singular values of $A$ are always well-conditioned, in the sense that changing $A$ in norm by at most $\epsilon$ can change any eigenvalue by at most $\epsilon$.

This is adequate for most purposes, unless the user is interested in the leading digits of a small singular value, one less than or equal to $\epsilon$ in magnitude. For example, computing $\sigma_i = 10^{-5}$ to within plus or minus $\epsilon = 10^{-4}$ means that no leading digits of the computed $\sigma_i$ may be correct. See [114,118] on the discussion of the sensitivity of small singular values and of when their leading digits may be computed accurately.

Singular vectors and singular subspaces, on the other hand, can be ill-conditioned. The example in §2.2.5 illustrates this point. (The eigenvalues and eigenvectors of that matrix are identical to its singular values and singular vectors.) Thus, the condition number of a singular vector depends on the gap between its singular value and the closest other singular value; the smaller the gap, the more sensitive the singular vector. When singular vectors corresponding to a cluster of close singular values are too ill-conditioned, the user may want to compute a basis of the singular subspace they span instead of individual singular vectors.