Numerical Algorithms

, Volume 70, Issue 3, pp 469–484 | Cite as

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

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


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.


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

Mathematical Subject Classifications (2010)

65F18 15A48 37N30 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Kanae Akaiwa
    • 1
    Email author
  • 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

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