Bipartite graphs are used to describe the generalized Schur complements of real matrices having nos quare 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 dagger may not be uniquely determined by the sign pattern of A. Analogous results are shown to hold for zero patterns.
|Journal||Electronic Journal of Linear Algebra|
|Publication status||Published - 2005|
- nearly reducible matrix
- bipartite graph
- LU factorization
- minimally strongly connected digraph.
- sign pattern
- zero pattern
- Schur complement