ELA, Volume 14, pp. 2-11, January 2005, abstract.

Schur complements of matrices with acyclic 
bipartite graphs

T. Britz, D.D. Olesky, and P. van den Driessche

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 the 
inverse of L and, if A is nonsingular, of the inverse 
of U. However, if A is singular, then the sign pattern 
of the Moore-Penrose inverse of U may not be uniquely 
determined by the sign pattern of A. Analogous results 
are shown to hold for zero patterns.