Max-algebra and pairwise comparison matrices, II |
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Authors: | L Elsner P van den Driessche |
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Institution: | a Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany b Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4 |
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Abstract: | This paper is a continuation of our 2004 paper “Max-algebra and pairwise comparison matrices”, in which the max-eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix. This approximation was based on minimizing the maximal relative error. In a later paper by Dahl a different error measure was used and led to a slightly different approximating transitive matrix. Here some geometric properties of this approximation problem are discussed. These lead, among other results, to a new characterization of a max-eigenvector of an irreducible nonnegative matrix. The case of Toeplitz matrices is discussed in detail, and an application to music theory that uses Toeplitz symmetrically reciprocal matrices is given. |
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Keywords: | Max-eigenvector SR-matrix Perturbations Toeplitz matrix Music theory |
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