TY - JOUR

T1 - Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

AU - Watanabe, Sennosuke

AU - Fukuda, Akiko

AU - Shigitani, Hitomi

AU - Iwasaki, Masashi

N1 - Funding Information:
This work was partially supported by Grants-in-Aid for Young Scientist (B) No. 16K21368 of the Japan Society for the Promotion of Science.

PY - 2018/10/8

Y1 - 2018/10/8

N2 - 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.

AB - 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.

KW - eigenvalue

KW - min-plus algebra

KW - ultradiscrete Toda equation

KW - weighted directed graph

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U2 - 10.1088/1751-8121/aae325

DO - 10.1088/1751-8121/aae325

M3 - Article

AN - SCOPUS:85054848984

VL - 51

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 44

M1 - 444001

ER -