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


    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
    Pages (from-to)2-11
    Publication statusPublished - 2005


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


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