Given a symmetric matrix
X, we consider the problem of finding a low-rank positive approximant of
X. That is, a symmetric positive semidefinite matrix,
S, whose rank is smaller than a given positive integer,
, which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten
p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.