Geodesic
Schur complements of matrices with acyclic bipartite graphs
The electronic journal of linear algebra, Tome 14 (2005), pp. 2-11.

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

Summary: Bipartite graphs are used to describe the generalized Schur complements of real matrices having no square submatrix with two or more nonzero diagonals. For any matrix A with this property, including any nearly reducible matrix, the sign pattern of each generalized Schur complement is shown to be determined uniquely by the sign pattern of A. Moreover, if A has a normalized LU factorization A = LU, then the sign pattern of A is shown to determine uniquely the sign patterns of L and U, and (with the standard LU factorization) of L-1 and, if A is nonsingular, of U-1. However, if A is singular, then the sign pattern of the Moore-Penrose inverse U# may not be uniquely determined by the sign pattern of A. Analogous results are shown to hold for zero patterns.
Classification : 05C50, 15A09, 15A23
Keywords: Schur complement, LU factorization, bipartite graph, sign pattern, zero pattern, nearly reducible matrix, minimally strongly connected digraph 
@article{ELA_2005__14__a4,
     author = {Britz, T. and Olesky, D.D. and van den Driessche, P.},
     title = {Schur complements of matrices with acyclic bipartite graphs},
     journal = {The electronic journal of linear algebra},
     pages = {2--11},
     publisher = {mathdoc},
     volume = {14},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2005__14__a4/}
}
TY  - JOUR
AU  - Britz, T.
AU  - Olesky, D.D.
AU  - van den Driessche, P.
TI  - Schur complements of matrices with acyclic bipartite graphs
JO  - The electronic journal of linear algebra
PY  - 2005
SP  - 2
EP  - 11
VL  - 14
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ELA_2005__14__a4/
LA  - en
ID  - ELA_2005__14__a4
ER  - 
%0 Journal Article
%A Britz, T.
%A Olesky, D.D.
%A van den Driessche, P.
%T Schur complements of matrices with acyclic bipartite graphs
%J The electronic journal of linear algebra
%D 2005
%P 2-11
%V 14
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ELA_2005__14__a4/
%G en
%F ELA_2005__14__a4
Britz, T.; Olesky, D.D.; van den Driessche, P. Schur complements of matrices with acyclic bipartite graphs. The electronic journal of linear algebra, Tome 14 (2005), pp. 2-11. http://geodesic.mathdoc.fr/item/ELA_2005__14__a4/