Eigenvalue Problem in Min-Plus Algebra

概要

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In the present paper, we focus on the eigenvalue problem of matrices with entries in min-plus algebra. Min-plus algebra is one of many idempotent semirings which have been studied in various fields of mathematics. Many of the theorems and techniques that we use in classical linear algebra seems to have analogues in linear algebra over min-plus algebra. However, such kinds of investigation have not yet exploited sufficiently. In the present paper, we try to give a characterization of eigenvalues of matrices with entries in min-plus algebra. We prove that eigenvalues of the weighted adjacency matrix of the network on digraphs with entries in min-plus algebra correspond to the average weight of the directed circuits. However, it is impossible for us to prove that there always exists an eigenvalue of weighted adjacency matrices for an arbitrary circuit of the network. So, the problem to characterize circuits for which there exits eigenvalues that are equal to the average weight of the circuit is left as an open problem.