Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

Sennosuke Watanabe, Akiko Fukuda, Hitomi Shigitani, Masashi Iwasaki

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: ⊕:= min and ⊗ := +. We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.

Original languageEnglish
Article number444001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number44
Publication statusPublished - 2018 Oct 8


  • eigenvalue
  • min-plus algebra
  • ultradiscrete Toda equation
  • weighted directed graph

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)


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