%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 <em>X</em>, we consider the problem of finding a low-rank positive approximant of <em>X</em>. That is, a symmetric positive semidefinite matrix, <em>S</em>, whose rank is smaller than a given positive integer, <img src=\"Edit_4fd853e4-6c10-4b35-adee-1eb354583ffa.bmp\" alt=\"\" height=\"15\" width=\"5\" />, 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 <em>p</em>-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