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

TL;DR

This paper characterizes the null space and rank of acyclic matrices with zero diagonal entries by relating them to the adjacency matrix of a forest, and provides an efficient algorithm for finding sparsest null space bases.

Contribution

It introduces a method to derive null space and rank of such matrices from forest adjacency matrices and offers an optimal algorithm for sparsest null space basis computation.

Findings

Null space and rank are obtained via non-singular diagonal matrices.

Provides an optimal time algorithm for sparsest null space basis.

Connects acyclic matrices with forest adjacency matrices.

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 null space and the rank of a 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 non-singular diagonal matrices. We combine these methods 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.