Efficient diagonalization of symmetric matrices associated with graphs of small treewidth

Abstract

Let $M=\left(m_{ij}\right)$ be a symmetric matrix of order n whose elements lie in an arbitrary field $F$, and let G be the graph with vertex set $\{1,\dots ,n\}$ such that distinct vertices i and j are adjacent if and only if $m_{ij}\ne 0$. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to M. If G is given with a tree decomposition $T$ of width k, then this can be done in time $O\left(k\right|T|+k^{2}n)$, where $\left|T\right|$ denotes the number of nodes in $T$. Among other things, this allows the computation of the determinant, the rank and the inertia of a symmetric matrix in time $O\left(k\right|T|+k^{2}n)$.