P-matrices and signed digraphs

Banaji, Murad ORCID: https://orcid.org/0000-0002-4983-0377 and Rutherford, Carrie (2010) P-matrices and signed digraphs. Discrete Mathematics, 311 (4) . pp. 295-301. ISSN 0012-365X [Article] (doi:10.1016/j.disc.2010.10.018)

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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 P_0-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 P_0-matrix. The results generalise a number of previous results relating P- and P_0-matrices to graphs.

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