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

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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.

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Yoshida-Hommachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Kanae Akaiwa & Yoshimasa Nakamura

  2. Department of Informatics and Environmental Science, Kyoto Prefectural University, 1-5 Nakaragi-cho, Shimogamo, Sakyo-ku, Kyoto, 606-8522, Japan

    Masashi Iwasaki

  3. Graduate School of Science and Engineering, Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe City, Kyoto, 610-0394, Japan

    Hisayoshi Tsutsumi & Koichi Kondo

Authors
  1. Kanae Akaiwa
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  2. Yoshimasa Nakamura
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  3. Masashi Iwasaki
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  4. Hisayoshi Tsutsumi
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  5. Koichi Kondo
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Correspondence to Kanae Akaiwa.

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Akaiwa, K., Nakamura, Y., Iwasaki, M. et al. A finite-step construction of totally nonnegative matrices with specified eigenvalues. Numer Algor 70, 469–484 (2015). https://doi.org/10.1007/s11075-015-9957-x

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  • DOI: https://doi.org/10.1007/s11075-015-9957-x

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