P-matrices and signed digraphs

TL;DR

This paper explores the relationship between signed digraphs and matrix products, showing how cycle parity in the graph influences the positivity properties of the resulting matrices, extending previous graph-matrix theory results.

Contribution

It establishes a novel link between cycle parity in signed digraphs and the P0-matrix property of matrix products, generalizing earlier findings.

Findings

Absence of e-cycles implies the matrix product is P0

Presence of e-cycle can produce a non-P0 matrix product

Generalizes previous graph-matrix relationship results

Abstract

We associate a signed digraph with a list of matrices whose dimensions permit them to be multiplied, and whose product is square. Cycles in this graph have a parity, that is, they are either even (termed e-cycles) or odd (termed o-cycles). The absence of e-cycles in the graph is shown to imply that the matrix product is a P0-matrix, i.e., all of its principal minors are nonnegative. Conversely, the presence of an e-cycle is shown to imply that there exists a list of matrices associated with the graph whose product fails to be a P0-matrix. The results generalise a number of previous results relating P- and P0-matrices to graphs.