ELA, Volume 11, pp. 115-131, June 2004, abstract.

Matrix Inversion and Digraphs: the One Factor Case

Thomas Britz, D. Dale Olesky, and Pauline van den Driessche

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. Nonsingular 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 sign-nonsingular. Several 
conditions that are equivalent to the inverse of a 
nonsingular nearly reducible matrix being nearly reducible 
are obtained.