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.
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The authors thank the reviewers for their careful reading and insightful suggestions. This work is supposed by the Grant-in-Aid for JSPS Fellows No. 26 ⋅6045 and the Grant-in-Aid for Scientific Research (C) No. 26400208 from the Japan Society for the Promotion of Science.
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Akaiwa, K., Nakamura, Y., Iwasaki, M. et al. An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues. Numer Algor 75, 1079–1101 (2017). https://doi.org/10.1007/s11075-016-0231-7
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DOI: https://doi.org/10.1007/s11075-016-0231-7
Keywords
- Extended discrete hungry Toda equation
- Finite-step construction
- Inverse eigenvalue problem
- Totally nonnegative