  This is a systematic investigation into the sensitivity of low-rank
approximations of real matrices. We show that the low-rank approximation
errors, in the two-norm, Frobenius norm and more generally, any Schatten
p-norm, are insensitive to additive rank-preserving perturbations in the
projector basis; and to matrix perturbations that are additive or change the
number of columns (including multiplicative perturbations). Thus, low-rank
matrix approximations are always well-posed and do not require a singular value
gap. In the presence of a singular value gap, connections are established
between low-rank approximations and subspace angles.
