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Schur complements of matrices with acyclic bipartite graphs. (English) Zbl 1103.05048
Authors’ abstract: Bipartite graphs are used to describe the generalized Schur completeness 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 \(L^{-1}\) 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.
MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A09 Theory of matrix inversion and generalized inverses
15A23 Factorization of matrices
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