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.