  When a projection of a symmetric or Hermitian matrix to the positive
semidefinite cone is computed approximately (or to working precision on a
computer), a natural question is to quantify its accuracy. A straightforward
bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and
Weyl bounds) suggests that the accuracy would be inversely proportional to the
spectral gap, implying it can be poor in the presence of small eigenvalues.
This work shows that a small gap is not a concern for projection onto the
semidefinite cone, by deriving error bounds that are gap-independent.
