Matrix inversion and digraphs: the one factor case

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Thomas Britz
Dale D. Olesky
Pauline van den Driessche

Abstract

The novel concept of a cyclic sequence of a digraph that has precisely one factor is defined, and is used to characterize the entries of the inverse of a matrix with such a digraph. This leads to a characterization of a strongly sign-nonsingular matrix in terms of cyclic sequences. Non-singular nearly reducible matrices are a well-known class of matrices having precisely one nonzero diagonal, and a simple expression for the entries of the inverse of such a matrix in terms of cyclic sequences is derived. A consequence is that a nonsingular nearly reducible matrix is strongly signnonsingular. Several conditions that are equivalent to the inverse of a nonsingular nearly reducible matrix being nearly reducible are obtained. 

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