New developments of matrix similarity transformations derived from integrable systems

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kyoto Prefectural University
• Principal Investigator: Iwasaki Masashi 京都府立大学, 生命環境科学研究科, 准教授 (30397575)
• Co-Investigator(Renkei-kenkyūsha): YAMAMOTO Yusaku 電気通信大学, 大学院情報理工学研究科, 教授 (20362288) ; ISHIWATA Emiko 東京理科大学, 理学部, 教授 (30287958) ; KONDO Koichi 同志社大学, 理工学部, 教授 (30314397) ; FUKUDA Akiko 芝浦工業大学, システム理工学部, 准教授 (70609297)
• Research Collaborator: SHINJO Masato
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 行列の相似変形 / dLVsアルゴリズム / 分割型ツイスト分解法 / 離散ハングリー可積分系 / 行列の帯構造 / フィボナッチ数列 / 逆固有値問題 / min-plus固有多項式

Outline of Final Research Achievements

One of the results is to improve the I-SVD algorithm for computing singular value decompositions of bidiagonal matrices with higher accuracy. The new I-SVD algorithm employs the proposed dLVs algorithm for singular values and the proposed divided Twisted factorization method for singular vectors. The second is to investigate the convergence of solutions to dynamical systems to eigenvalues of various band matrices. Solution expressions of the discrete hungry integrable systems and their asymptotic convergence are thoroughly clarified. A new algorithm for computing eigenvalues of pentadiagonal matrices is also designed. The third is to find relationships among the discrete hungry integrable systems, extended Fibonacci sequences and roots of polynomials, and then develop them in constructing band matrices with prescribed eigenvalues. New characteristic polynomials of matrices are also presented over min-plus algebra.

Free Research Field

行列の固有値問題および逆固有値問題、離散可積分系