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

Daniel A. Jaume; Adrián Pastine

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arXiv · 1802.10142

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

Daniel A. Jaume·Adrián Pastine

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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.

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Daniel A. Jaume, Adrián Pastine. 2018-02-27. On the structure of the fundamental subspaces of acyclic matrices with $0$ in the diagonal. https://arxiv.org/abs/1802.10142

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