A max version of the perron-frobenius theorem

Bapat, R. B. (1998) A max version of the perron-frobenius theorem Linear Algebra and its Applications, 275-276 . pp. 3-18. ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00243...

Related URL: http://dx.doi.org/10.1016/S0024-3795(97)10057-X


If A an n × n nonnegative, irreducible matrix, then there exists μ(A) > 0, and a positive vector x such that maxjaijxj= μ(A)xi, i = 1, 2,..., n. Furthermore, μ(A) is the maximum geometric mean of a circuit in the weighted directed graph corresponding to A. This theorem, which we refer to as the max version of the Perron-Frobenius Theorem, is well-known in the context of matrices over the max algebra and also in the context of matrix scalings. In the present work, which is partly expository, we bring out the intimate connection between this result and the Perron-Frobenius theory. We present several proofs of the result, some of which use the Perron-Frobenius Theorem. Structure of max eigenvalues and max eigenvectors is described. Possible ways to unify the Perron-Frobenius Theorem and its max version are indicated. Some inequalities for μ(A) are proved.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Max algebra; Nonnegative Matrix; Perron-Frobenius Theorem
ID Code:1382
Deposited On:05 Oct 2010 12:40
Last Modified:13 May 2011 08:11

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