Spectral bounds for certain special type of rational matrices

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.