Gauss elimination without pivoting for positive semidefinite matrices and an application to sum of squares representations

Abstract

It is shown that Gauss elimination without pivoting is possible for positive semidefinite matrices. While we do not claim the method as numerically the most advisable, it allows to obtain sum of squares (sos) representations in a more direct way and with more theoretical insight, than by the usual text book proposals. The result extends a theorem attributed for definite quadratic forms to Lagrange and Beltrami and is useful as a finishing step in recent algorithms by Powers and WöNormann [PW] and Parillo [PSPP] to write polynomials p ¸ IR[x] = IR[x1, ..., xn] as a sum of squares in IR[x] when such a representation exists.