Generalized inverses of symmetric M -matrices

In this work we carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M -matrices. The key idea in our approach is to identify any symmetric M -matrix with a positive semi-definite Schrödinger operator on a connected network whose conductances are given by th...

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Bibliographic Details
Published in:Linear algebra and its applications 2010-04, Vol.432 (9), p.2438-2454
Main Authors: Bendito, E., Carmona, A., Encinas, A.M., Mitjana, M.
Format: Article
Language:English
Subjects:
[formula omitted]-matrix
Algebra
Circulant matrices
Equilibrium measure
Exact sciences and technology
Generalized inverses
Linear and multilinear algebra, matrix theory
Mathematics
Schrödinger operator
Sciences and techniques of general use
Triangular matrices
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Summary:In this work we carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M -matrices. The key idea in our approach is to identify any symmetric M -matrix with a positive semi-definite Schrödinger operator on a connected network whose conductances are given by the off-diagonal elements of the M -matrix. Moreover, the potential of the operator is determined by the positive eigenvector of the M -matrix. We prove that any generalized inverse can be obtained throughout a Green kernel plus some projection operators related to the positive eigenfunction. Moreover, we use the discrete Potential Theory associated with any positive semi-definite Schrödinger operator to get an explicit expression for any generalized inverse, in terms of equilibrium measures. Finally, we particularize the obtained result to the cases of tridiagonal matrices and circulant matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2009.11.008