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Let $M=(m_{ij})$ be a symmetric matrix of order $n$ whose elements lie in anarbitrary field $\mathbb{F}$, and let $G$ be the graph with vertex set$\{1,\ldots,n\}$ such that distinct vertices $i$ and $j$ are adjacent if andonly if $m_{ij} \neq 0$. We introduce a dynamic programming algorithm thatfinds a diagonal matrix that is congruent to $M$. If $G$ is given with a treedecomposition $\mathcal{T}$ of width $k$, then this can be done in time$O(k|\mathcal{T}| + k^2 n)$, where $|\mathcal{T}|$ denotes the number of nodesin $\mathcal{T}$. Among other things, this allows one to compute thedeterminant, the rank and the inertia of a symmetric matrix in time$O(k|\mathcal{T}| + k^2 n)$.

Efficient diagonalization of symmetric matrices associated with graphs of small treewidth