# zbMATH — the first resource for mathematics

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
Full Text: