Estimation of bounds for the zeros of a polynomial using numerical radius
Let \(\lambda\) be a zero of the polynomial \(p(z)=z^n+a_{n-1}z^{n-1}+\dots +a_1+a_0\), where \(a_0,\dots, a_{n-1} \in \mathbb C\). The authors estimate the upper bound of \(\lambda\) using Hilbert space techniques involving the Frobenius companion matrix and the numerical radius. They start with the result that for a matrix \(A\in M_n(\mathbb C)\) such that \(A^2-(\lambda_1+\lambda_2)A+\lambda_1 \lambda_2 I=0\), \(\lambda_1, \lambda_2\in \mathbb C\), the numerical range \(W(A)=\{\langle Ax,x\rangle :\,\, \| x\| =1\}\) is an ellipse with foci at \(\lambda_1\) and \(\lambda_2\). The axes and the inclination of the ellipse are described. In the case \(A=\left( \begin{matrix} \lambda_1 I & B\\ 0 & \lambda_2 I \end{matrix} \right) \), the numerical radius is \(w(A)=\sqrt{x_0^2+y_0^2}\). The parameters \(x_0\) and \(y_0\) are given. Using the above results, the authors get bounds for \(\lambda\). Interesting examples illustrate the main results.
scientific article; zbMATH DE number 6535254