Schur complements of matrices with acyclic bipartite graphs

Thomas Johann Britz, D.D. Olesky, P. van den Driessche

    Research output: Contribution to journalJournal articleResearchpeer-review

    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 languageEnglish
    JournalElectronic Journal of Linear Algebra
    Volume14
    Pages (from-to)2-11
    ISSN1081-3810
    Publication statusPublished - 2005

    Keywords

    • nearly reducible matrix
    • bipartite graph
    • LU factorization
    • minimally strongly connected digraph.
    • sign pattern
    • zero pattern
    • Schur complement

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