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.