Bounds on the moduli of eigenvalues of rational matrices

Abstract

A rational matrix is a matrix-valued function $R(\lambda ):\mathbb{C}\to M_p$ such that $R(\lambda )={\left[\begin{array}{c}{r}_{ij}(\lambda )\end{array}\right]}_{p\times p}$, where $r_{ij}(\lambda )$ are scalar complex rational functions in $\lambda$ for $i,j=1,2,\dots ,p$. The aim of this paper is to obtain bounds on the moduli of eigenvalues of rational matrices in terms of the moduli of their poles. To a given rational matrix $R(\lambda )$ we associate a block matrix ${\mathcal{C}}_R$ whose blocks consist of the coefficient matrices of $R(\lambda )$, as well as a scalar real rational function $q(x)$ whose coefficients consist of the norm of the coefficient matrices of $R(\lambda )$. We prove that a zero of $q(x)$ which is greater than the moduli of all the poles of $R(\lambda )$ will be an upper bound on the moduli of eigenvalues of $R(\lambda )$. Moreover, by using a block matrix associated with $q(x)$, we establish bounds on the zeros of $q(x)$, which in turn yields bounds on the moduli of eigenvalues of $R(\lambda )$.

arXiv commentA few grammatical mistakes have been corrected. Two new references have been added