Abstract
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.
Original language | English |
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Journal | Electronic Journal of Linear Algebra |
Volume | 14 |
Pages (from-to) | 2-11 |
ISSN | 1081-3810 |
Publication status | Published - 2005 |
Keywords
- nearly reducible matrix
- bipartite graph
- LU factorization
- minimally strongly connected digraph.
- sign pattern
- zero pattern
- Schur complement