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Bipartite graphs and inverse sign patterns of strong sign-nonsingular matrices. (English) Zbl 0807.05053
Authors’ abstract: A sign-nonsingular matrix is a square matrix \(X\) such that each matrix \(Y\) with the same sign pattern as \(X\) is nonsingular. If, in addition, the sign pattern of the inverse of \(Y\) is the same for all \(Y\), then \(X\) is a strong sign-nonsingular matrix. A fully indecomposable matrix is a matrix whose associated bipartite graph is connected and (perfect) matching covered. The bipartite graphs of fully indecomposable, strong sign-nonsingular matrices are characterized and a recursive construction is given. This characterization is used to determine the sign patterns of the inverses of fully indecomposable, strong sign-nonsingular matrices and to develop a recognition algorithm for such sign patterns. Those maximal strong sign-nonsingular matrices whose sign patterns are uniquely determined by the sign patterns of their inverses are also characterized in terms of bipartite graphs.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A09 Theory of matrix inversion and generalized inverses
05C75 Structural characterization of families of graphs
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