On the structure of the fundamental subspaces of acyclic matrices with 0 in the diagonal

Abstract

A matrix is called acyclic if replacing the diagonal entries with \(0\), and the nonzero diagonal entries with \(1\), yields the adjacency matrix of a forest. In this paper we show that the null space and the rank of an acyclic matrix with \(0\) in the diagonal is obtained from the null space and the rank of the adjacency matrix of the forest by multipliying by nonsingular diagonal matrices. We combine these with an algorithm for finding a sparsest basis of the null space of a forest to provide an optimal time algorithm for finding a sparsest basis of the null space of acyclic matrices with \(0\) in the diagonal.