Generalized inverse-positivity and splittings of M-matrices |
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Authors: | M Neumann RJ Plemmons |
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Institution: | Department of Mathematics University of Nottingham Nottingham NG7 2RD, England;Department of Computer Science and Mathematics University of Tennessee Knoxville, Tennessee 37916 USA |
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Abstract: | The concepts of matrix monotonicity, generalized inverse-positivity and splittings are investigated and are used to characterize the class of all M-matrices A, extending the well-known property that A?1?0 whenever A is nonsingular. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. It is shown how the nonnegativity of a generalized left inverse of A plays a fundamental role in such characterizations, thereby extending recent work by one of the authors, by Meyer and Stadelmaier and by Rothblum. In addition, new characterizations are provided for the class of M-matrices with “property c”; that is, matrices A having a representation A=sI?B, s>0, B?0, where the powers of converge. Applications of these results to the study of iterative methods for solving arbitrary systems of linear equations are given elsewhere. |
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