Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation. (8th October 2018)

• Record Type: Journal Article
• Authors: Watanabe, Sennosuke; Fukuda, Akiko; Shigitani, Hitomi; Iwasaki, Masashi
• Is Part Of: Journal of physics. Volume 51:Number 44(2018)
• Publication Date: 2018-10-08
• Subjects: eigenvalue -- min-plus algebra -- ultradiscrete Toda equation -- weighted directed graph; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aae325
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: 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: and . We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.