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

Abstract

A complex square matrix is called a ray nonsingular matrix (RNS matrix) if its ray pattern implies that it is nonsingular. A matrix $M=I-A\left(W\right)$ 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 $M=I-A\left(W\right)$ is a forbidden structure for RNS if and only if $M$ is not RNS.

Introduction

The ray of a complex number $z$ is defined to be

$$ray\left(z\right)=\left\{\begin{array}{cc}z∕\left|z\right|,\hfill & z\ne 0;\hfill \\ 0,\hfill & z=0.\hfill \end{array}\right.$$

The ray pattern of a complex matrix $A$, denoted by $ray\left(A\right)$, is the matrix obtained from $A$ by replacing each of its entries with the corresponding rays. The set of complex matrices with the same ray pattern as $A$ is called the ray pattern class of $A$, denoted by $Q_R\left(A\right)$, i.e., 

$$Q_R\left(A\right)=\left\{B\right|ray\left(B\right)=ray\left(A\right)\}.$$

If $A=ray\left(A\right)$, then $A$ is called a ray pattern matrix.

As a natural generalization of sign pattern matrices (see [2], [6]) from the real case to the complex case, the study of ray pattern matrices has received considerable attentions during the last two decades. For example, in [5], [16], [18] and related articles, the spectrally arbitrary ray patterns are studied; in [4], [8], [19] the powers of ray pattern matrices is studied; in [17] the ray solvability of complex linear system is studied, and the Moore–Penrose inverse of ray pattern matrices are studied in [3].

In this paper, we are interested in the ray nonsingular matrices (abbreviated as  RNS matrices). A complex square matrix $A$ is called ray nonsingular if for any matrix $B$ with $ray\left(B\right)=ray\left(A\right)$, $B$ is nonsingular. In other words, RNS matrices are the matrices that can be confirmed to be nonsingular when only the ray patterns of the matrices are provided. Throughout this paper, we will only talk about complex square matrices and concern whether or not they are RNS matrices.

RNS matrices are a generalization of the well known SNS matrices, which is a fundamental concept in the study of sign solvable linear systems (see [2] for detail). The concept was introduced together with the introduction of ray patterns in [15]. A sufficient condition of ray nonsingularity was also given in [15] using the sets of signed transversal products, and an example was given to show that this sufficient condition is not necessary for the partly decomposable case. In [7], a matrix was given to show that the sufficient condition is also not necessary for the fully indecomposable case. Also in [7], an example of 4 × 4 full RNS matrices was given, and it was proved that there exist no full RNS matrices whose orders are at least 6. The non-existence of 5 × 5 full RNS matrices was proved in [9]. In addition, a special type of ray nonsingular matrices called inverse closed ray nonsingular matrices were studied in [10].

The problem of how to recognize RNS matrices is still open. In [13], an algorithm is given to determine whether or not a given cycle chain matrix is RNS. In [12] the algorithm was generalized to cycle tree matrices.

SNS matrices are characterized by using the language of forbidden structures. It is shown that positive cycles are the only type of minimum forbidden structures for the sign nonsingular property in [1] (with a precondition that all the diagonal entries are negative). In [14], it is shown that a cycle chain matrix $M$ that is not RNS is a forbidden structure for RNS if and only if it is not an odd cycle chain matrix.

In this paper, we will show that for the cycle tree matrix $M=I-A\left(W\right)$, if there is no positive cycles in $W$, then $M$ is a forbidden structure for RNS if and only if $M$ is not RNS.