Numerical Algorithms

, Volume 70, Issue 3, pp 469–484

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

• Kanae Akaiwa
• Yoshimasa Nakamura
• Masashi Iwasaki
• Hisayoshi Tsutsumi
• Koichi Kondo
Original Paper

DOI: 10.1007/s11075-015-9957-x

Akaiwa, K., Nakamura, Y., Iwasaki, M. et al. Numer Algor (2015) 70: 469. doi:10.1007/s11075-015-9957-x

## Abstract

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.

### Keywords

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

### Mathematical Subject Classifications (2010)

65F18 15A48 37N30

## Authors and Affiliations

• Kanae Akaiwa
• 1
• Yoshimasa Nakamura
• 1
• Masashi Iwasaki
• 2
• Hisayoshi Tsutsumi
• 3
• Koichi Kondo
• 3
1. 1.Graduate School of InformaticsKyoto UniversitySakyo-kuJapan
2. 2.Department of Informatics and Environmental ScienceKyoto Prefectural UniversitySakyo-kuJapan
3. 3.Graduate School of Science and EngineeringDoshisha UniversityKyotanabe CityJapan