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SIAM J. MATRIX ANAL. APPL. c 2008 Society for Industrial and Applied Mathematics
Vol. 30, No. 2, pp. 731740
ON THE MINIMUM RANK AMONG POSITIVE SEMIDEFINITE
MATRICES WITH A GIVEN GRAPH
MATTHEW BOOTH, PHILIP HACKNEY, BENJAMIN HARRIS§, CHARLES R.
JOHNSON¶, MARGARET LAY , LON H. MITCHELL# , SIVARAM K. NARAYAN ,
AMANDA PASCOE , KELLY STEINMETZ§§ , BRIAN D. SUTTON¶¶ , AND WENDY
WANG
Abstract. Let P(G) be the set of all positive semidefinite matrices whose graph is G, and
msr(G) be the minimum rank of all matrices in P(G). Upper and lower bounds for msr(G) are given
and used to determine msr(G) for some wellknown graphs, including chordal graphs, and for all
simple graphs on less than seven vertices.
Key words. rank, positive semidefinite, graph of a matrix
AMS subject classifications. 15A18, 15A57, 05C50
DOI. 10.1137/050629793
1. Introduction. If A is an nbyn Hermitian matrix, then its graph G(A) is
the undirected, simple graph on vertices {1, 2, . . . , n}, which has an edge between
vertices i and j if and only if the i, j entry of A is nonzero and i = j. The graph is
independent of the real diagonal entries of A. The set of all Hermitian matrices that
