Bounding the zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition

TL;DR

This paper introduces new inequalities for the numerical radius of block matrices and applies them to effectively bound polynomial zeros using a partitioned Frobenius companion matrix, demonstrating improved results.

Contribution

It presents novel inequalities for block matrix numerical radius and applies them to enhance polynomial zero bounding via a partitioned Frobenius companion matrix.

Findings

New inequalities for the numerical radius of block matrices.

Effective polynomial zero bounds using partitioned Frobenius matrices.

Numerical example shows improved bounds over existing methods.

Abstract

In this work, Some new inequalities for the numerical radius of block $n$-by-$n$ matrices are presented. As an application, bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition approach is proved and affirmed by a numerical example showing that our approach of bounding zeros of polynomials could be very effective in comparison with the most famous and some recent results presented in the field.