An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues
An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues
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具有规定特征值的全非负矩阵的任意能带结构构造
DOI：
10.1007/s11075-016-0231-7
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Koichi Kondo
中科院分区：
文献类型：
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作者：
Kanae Akaiwa;Yoshimasa Nakamura;Masashi Iwasaki;Akira Yoshida;Koichi Kondo
具有规定特征值的全非负（TN）矩阵的构造是实值非负特征值反问题中的一个重要课题。TN矩阵是方阵，其子阵都是非负的。我们之前的论文(编号。Algor, 70, 469-484, ?？2015)基于可积系统研究中导出的离散饥饿Toda （dhToda）方程，提出了TN矩阵的有限步构造，该构造仅限于具有规定特征值的上或下Hessenberg形式。在我们之前的论文的基础上，我们构造了具有任意数目的上下三角形部分对角线和规定特征值的带状TN矩阵，涉及具有规定特征值的上Hessenberg，下Hessenberg和密集TN矩阵。我们首先准备了一个无限序列，它与不同的特征值和两个整数mandn相关联，它们分别决定了m-带矩阵的上带宽和下带宽。这两种人都在实现我们的目标方面发挥着关键作用。这项研究遵循了与我们之前的论文类似的思路，但由于引入了n而变得复杂。我们接下来考虑扩展的Hankel行列式和扩展的Hadamard多项式，涉及无穷序列的元素，然后推导它们之间的关系。这些关系帮助我们从dhToda方程的扩展角度理解带特征值的带状TN矩阵。最后，我们提出了一个有限步构造具有任意上下带宽和规定特征值的TN矩阵的方法，并给出了举例说明。
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 eigenvaluesand two integersMandNwhich determine the upper and lower bandwidths ofm-by-mbanded matrices, respectively. BothMandNplay a key role for achieving our purpose. The study follows similar lines to our previous paper, but is complicated by the introduction ofN. 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 eigenvaluesfrom 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.