Forbidden structures for ray nonsingularity among cycle tree matrices without positive cycles

摘要

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. A matrix M = I -A(W) is called a cycle tree matrix if the adjacency structure of the cycles in the arc-weighted digraph W (with no multi-arcs or loops), which is described by the cycle graph of W, is a tree. In this paper, it is shown that if there is no positive cycle in W, then the cycle tree matrix M = I - A(W) is a forbidden structure for RNS if and only if M is not RNS. (C) 2020 Elsevier B.V. All rights reserved.