Spectral bounds for certain special type of rational matrices

TL;DR

This paper derives bounds on the eigenvalues of a special class of rational matrices using matrix analysis techniques, providing theoretical bounds and numerical illustrations for these bounds.

Contribution

It introduces new spectral bounds for rational matrices of a specific form, combining Bauer-Fike, Rouché's theorem, and numerical radius inequalities.

Findings

Upper bounds via Bauer-Fike theorem

Lower bounds using Rouché's theorem for matrix functions

Numerical examples illustrating bounds when coefficients are unitary

Abstract

The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(\lambda) = \displaystyle -B_0 +I\lambda +\frac{B_1}{\lambda-\alpha_1}+ \dots+ \frac{B_m}{\lambda-\alpha_m}$, where $B_i$'s are $n \times n$ complex matrices and $\alpha_i$'s are distinct complex numbers, using the following methods: $(1)$ an upper bound is obtained using the Bauer-Fike theorem for complex matrices on an associated block matrix $C_T$ of the given rational matrix $T(\lambda)$, $(2)$ a lower bound is obtained in terms of a zero of a scalar real rational function $p(x)$ associated with $T(\lambda)$, using Rouch$\text{\'e}$'s theorem for matrix-valued functions and $(3)$ an upper bound is also obtained using a numerical radius inequality for a block matrix $C_q$ associated with another scalar real rational function $q(x)$ corresponding to $T(\lambda)$. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.