%0 Journal Article %T Low-Rank Positive Approximants of Symmetric Matrices %A Achiya Dax %J Advances in Linear Algebra & Matrix Theory %P 172-185 %@ 2165-3348 %D 2014 %I Scientific Research Publishing %R 10.4236/alamt.2014.43015 %X 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. %K Low-Rank Positive Approximants %K Unitarily Invariant Matrix Norms %U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=50062