On a parametrization of positive semidefinite matrices with zeros

Details On a parametrization of positive semidefinite matrices with zeros On a parametrization of positive semidefinite matrices with zeros

2010-01-19

arxiv.org/abs/1001.3195v2

SIAM J. Matrix Anal. Appl. 31 (2010), no. 5, 2665--2680

Mathematics

Algebraic Geometry

16 pages, 1 figure. New shorter proof of Lemma 5.6. Final version, to appear in SIAM Journal on Matrix Analysis and Applicatio

Scientific paper

We study a class of parametrizations of convex cones of positive semidefinite matrices with prescribed zeros. Each such cone corresponds to a graph whose non-edges determine the prescribed zeros. Each parametrization in this class is a polynomial map associated with a simplicial complex supported on cliques of the graph. The images of the maps are convex cones, and the maps can only be surjective onto the cone of zero-constrained positive semidefinite matrices when the associated graph is chordal and the simplicial complex is the clique complex of the graph. Our main result gives a semi-algebraic description of the image of the parametrizations for chordless cycles. The work is motivated by the fact that the considered maps correspond to Gaussian statistical models with hidden variables.

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