On the Hegland-Marti inverse eigenvalue problem
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 \(n=3\).
scientific article; zbMATH DE number 1237987