Spectral bounds for certain special type of rational matrices

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.