Geodesic
Group inverses of matrices with path graphs
The electronic journal of linear algebra, Tome 17 (2008), pp. 219-233.

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

Summary: A simple formula for the group inverse of a $2 \times 2$ block matrix with a bipartite digraph is given in terms of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of such matrices are given. An extension of the graph-theoretic description of the group inverse to singular matrices with tree graphs is conjectured.
Classification : 15A09, 05C50
Keywords: group inverse, tridiagonal matrix, tree graph, Moore-Penrose inverse, bipartite digraph 
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     author = {Catral, Minerva and Olesky, Dale D. and van den Driessche, Pauline},
     title = {Group inverses of matrices with path graphs},
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Catral, Minerva; Olesky, Dale D.; van den Driessche, Pauline. Group inverses of matrices with path graphs. The electronic journal of linear algebra, Tome 17 (2008), pp. 219-233. http://geodesic.mathdoc.fr/item/ELA_2008__17__a28/