Schur complements of matrices with acyclic bipartite graphs

Abstract

Bipartite graphs are used todescribe the generalized Schur complements of real matrices having nosquare submatrix with twoor more nonzerodiagonals. 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.