Numerical Algorithms

, Volume 75, Issue 4, pp 1079–1101

An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues

Original Paper

Abstract

The construction of totally nonnegative (TN) matrices with prescribed eigenvalues is an important topic in real-valued nonnegative inverse eigenvalue problems. TN matrices are square matrices whose minors are all nonnegative. Our previous paper (Numer. Algor. 70, 469–484, ??2015) presented a finite-step construction of TN matrices limited to upper or lower Hessenberg forms with prescribed eigenvalues, based on the discrete hungry Toda (dhToda) equation which is derived from the study of integrable systems. Building on our previous paper, we produce the construction of banded TN matrices with an arbitrary number of diagonals in both lower and upper triangular parts and prescribed eigenvalues, involving upper Hessenberg, lower Hessenberg, and dense TN matrices with prescribed eigenvalues. We first prepare an infinite sequence associated with distinct eigenvalues $$\lambda _{1},\lambda _{2},\dots ,\lambda _{m}$$ and two integers M and N which determine the upper and lower bandwidths of m-by-m banded matrices, respectively. Both M and N play a key role for achieving our purpose. The study follows similar lines to our previous paper, but is complicated by the introduction of N. We next consider extended Hankel determinants and extended Hadamard polynomials involving elements of the infinite sequence and then derive their relationships. These relationships help us understand banded TN matrices with eigenvalues $$\lambda _{1},\lambda _{2},\dots ,\lambda _{m}$$ from the viewpoint of an extension of the dhToda equation. Finally, we propose a finite-step procedure for constructing TN matrices with an arbitrary upper and lower bandwidths and prescribed eigenvalues and also give illustrative examples.

Keywords

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

Mathematics Subject Classifications (2010)

65F18 15A48 37N30

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© Springer Science+Business Media New York 2016

Authors and Affiliations

• Kanae Akaiwa
• 1
• 4
Email author
• Yoshimasa Nakamura
• 1
• Masashi Iwasaki
• 2
• Akira Yoshida
• 3
• Koichi Kondo
• 3
1. 1.Graduate School of InformaticsKyoto UniversityKyotoJapan
2. 2.Department of Informatics and Environmental ScienceKyoto Prefectural UniversityKyotoJapan
3. 3.Graduate School of Science and EngineeringDoshisha UniversityKyotanabe CityJapan
4. 4.Faculty of Computer Science and EngineeringKyoto Sangyo UniversityKyotoJapan

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