Abstract
For any symmetric matrix A over a field, it is shown there is a permutation matrix P such that PAP T has a factorization PAP T = LDL T where L is unit lower triangular and D is block diagonal with 1-by-1 or 2-by-2 blocks. In particular, for the adjacency matrix of a bipartite graph G , the vertices of G may be ordered so that adj G = LDL T where all diagonal blocks of D are 2-by-2. Furthermore, if G is a tree, this factorization may be done with integer matrices.
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