A finite-step construction of totally nonnegative matrices with specified eigenvalues

被引：3

作者：

Akaiwa, Kanae ^{[1
]}

Nakamura, Yoshimasa ^{[1
]}

Iwasaki, Masashi ^{[2
]}

Tsutsumi, Hisayoshi ^{[3
]}

Kondo, Koichi ^{[3
]}

机构：

[1] Kyoto Univ, Grad Sch Informat, Sakyo Ku, Kyoto 6068501, Japan

[2] Kyoto Prefectural Univ, Dept Informat & Environm Sci, Sakyo Ku, Kyoto 6068522, Japan

[3] Doshisha Univ, Grad Sch Sci & Engn, Kyotanabe, Kyoto 6100394, Japan

来源：

NUMERICAL ALGORITHMS | 2015年 / 70卷 / 03期

基金：

日本学术振兴会;

关键词：

Finite-step construction; Totally nonnegative; Inverse eigenvalue problem; Discrete hungry Toda equation; ALGORITHM; SYSTEMS;

D O I：

10.1007/s11075-015-9957-x

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Matrices where all minors are nonnegative are said to be totally nonnegative (TN) matrices. In the case of banded TN matrices, which can be expressed by products of several bidiagonal TN matrices, Fukuda et al. (Annal. Mat. Pura Appl. 192, 423-445, 2013) discussed the eigenvalue problem from the viewpoint of the discrete hungry Toda (dhToda) equation. The dhToda equation is a discrete integrable system associated with box and ball systems. In this paper, we consider an inverse eigenvalue problem for such banded TN matrices by examining the properties of the dhToda equation. This problem is a real-valued nonnegative inverse eigenvalue problem. First, we show the determinant solution to the dhToda equation with suitable boundary conditions. Next, we clarify the relationship between the characteristic polynomials of the banded TN matrices and the determinant solution. Finally, taking this relationship into account, we design a finite-step procedure for constructing banded TN matrices with specified eigenvalues. We also present an example to demonstrate this procedure.

引用

页码：469 / 484

页数：16

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