Geodesic
Matrix inversion and digraphs: the one factor case
The electronic journal of linear algebra, Tome 11 (2004), pp. 115-131.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 signnonsingular. Several conditions that are equivalent to the inverse of a nonsingular nearly reducible matrix being nearly reducible are obtained.
Classification : 05C50, 15A09, 05C20
Keywords: inverse matrix, digraph, sign pattern, strongly sign-nonsingular matrix, nearly reducible matrix, minimally strongly connected digraph 
@article{ELA_2004__11__a11,
     author = {Britz, T. and Olesky, D.D. and van den Driessche, P.},
     title = {Matrix inversion and digraphs: the one factor case},
     journal = {The electronic journal of linear algebra},
     pages = {115--131},
     publisher = {mathdoc},
     volume = {11},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2004__11__a11/}
}
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Britz, T.; Olesky, D.D.; van den Driessche, P. Matrix inversion and digraphs: the one factor case. The electronic journal of linear algebra, Tome 11 (2004), pp. 115-131. http://geodesic.mathdoc.fr/item/ELA_2004__11__a11/