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For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\\mathbb{F})$ be the set of all symmetric $n\\times n$ matrices $A=[a_{i,j}]$ over $\\mathbb{F}$ with $a_{i,j}\\not=0$, $i\\not=j$ if and only if $ij\\in E$. We show that if $G$ is the complement of a partial $k$-tree and $m\\geq k+2$, then for all nonsingular symmetric $m\\times m$ matrices $K$ over $\\mathbb{F}$, there exists an $m\\times n$ matrix $U$ such that $U^T K U\\in S(G;\\mathbb{F})$. 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