Derivation of new discrete integrable systems and its applications to inverse eigenvalue problems

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K18229
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics; Basic analysis
• Research Institution: Kyoto Sangyo University
• Principal Investigator: Akaiwa Kanae 京都産業大学, 情報理工学部, 准教授 (30771878)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 逆固有値問題 / 全非負行列 / 離散可積分系 / 直交多項式

Outline of Final Research Achievements

Inverse eigenvalue problems (IEPs) are problems to construct matrices with prescribed eigenvalues. Especially, it is difficult to solve IEPs for totally nonnegative (TN) matrices, where all minors are nonzero.
Nonlinear equations whose solutions are explicitly expressed are called integrable systems, and time-discretization of integrable systems are discrete integrable systems. One of typical discrete integrable systems is discrete Toda equation.
In this research, we proposed new algorithms to solve IEPs for a band TN matrix with an arbitrary bandwidth, and a TN matrix with zig-zag structure called Laurent-Jacobi matrix based on discrete two-dimensional Toda equation with a reduction condition, and discrete relativistic Toda equation, respectively.

Free Research Field

応用可積分系

Academic Significance and Societal Importance of the Research Achievements

逆固有値問題研究の多くは行列解析分野からのアプローチであり、ある固有値をもつ行列の性質や特定のサイズ・特徴についての研究が多い。本研究のように、指定した固有値をもつ行列を具体的に作成する手法は珍しいため、可積分系分野だけでなく行列解析分野へも貢献できる。近似解ではなく厳密解を有限回反復で求められることも大きな特色である。
提案手法はパラメータの条件を緩めればTN行列以外の行列も作成可能なため汎用性が高い。TN行列が現れる建築物の構造解析等の実問題はもちろんのこと、他の工学的・理学的問題への応用が期待される。