On the Hegland-Marti inverse eigenvalue problem
type: Item
instance of: scholarly article
title: On the Hegland-Marti inverse eigenvalue problem
zbMATH Open document ID: 0913.15008
DOI: 10.1007/BF02355014
author: K. Appert
published in: Mathematical Notes
publication date: 1999-06-08T00:00:00Z
review text: The additive inverse eigenvalue problem is stated as follows. Let \(A\) be a given \(n\times n\) matrix over a subfield \({\mathbf F}\) of the field \(\mathbb{C}\); without loss of generality, we can assume that the diagonal entries of \(A\) are zero. It is required to find a diagonal matrix \(D=\text{diag} (d_1, \dots, d_n) \in \mathbb{F}^{n,n}\) such that the spectrum of the matrix \(C= A+D\) is the prescribed set of numbers \(\lambda= (\lambda_1, \dots, \lambda_n\} \in\mathbb{C}\). Additive inverse problems are always solvable over \(\mathbb{C}\) and have finitely many solutions. An interesting special additive invers eigenvalue problem was considered by \textit{M. Hegland} and \textit{J. T. Marti} [SIAM J. Matrix Anal. Appl. 10, No. 2, 219-228 (1989; Zbl 0674.65015)]. The matrix of the Hegland-Marti problem has the form \[ A=T_n= \left( \begin{matrix} 0 & 1 & \dots & 0 & 0 \\ 1 & 0 & \dots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & \dots & 1 & 0 \end{matrix} \right). \] Hegland and Marti consider this problem over the real number field (assuming that the prescribed eigenvalues are also real numbers) and discuss some iterative numerical methods for solving the problem. We are interested in the question of whether or not Hegland and Marti had an alternative to the iterative approach and why they did not try to construct a direct (i.e., finite) algorithm only involving arithmetic operations and extracting roots. In order words: Can the Hegland-Marti problem be solved in radicals? Our question is motivated by the well-known fact that there exist inverse eigenvalue matrix problems that can be solved in radicals (and even in only quadratic radicals). In this work, we prove that the Hegland-Marti problem is unsolvable in radicals over \(\mathbb{Q}\) for \(n=4\), although it can be solved in radicals for \(n=2\) and \(n=3\).
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1237987
zbMATH Keywords: additive inverse eigenvalue problem
zbMATH Keywords: Hegland-Marti problem
zbMATH Keywords: solvability in radicals
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: Galois group
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02355014
OpenAlex ID: W2025766532
description: scientific article; zbMATH DE number 1237987