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.

翻译：本手稿旨在推导特殊类型有理矩阵 $T(\lambda) = \displaystyle -B_0 +I\lambda +\frac{B_1}{\lambda-\alpha_1}+ \dots+ \frac{B_m}{\lambda-\alpha_m}$ 特征值模的界，其中 $B_i$ 为 $n \times n$ 复矩阵，$\alpha_i$ 为互异复数。采用以下方法：(1) 对给定有理矩阵 $T(\lambda)$ 的关联分块矩阵 $C_T$ 应用复矩阵的 Bauer-Fike 定理，获得上界；(2) 结合矩阵值函数的 Rouch$\text{\'e}$ 定理，通过与 $T(\lambda)$ 关联的标量实有理函数 $p(x)$ 的零点建立下界；(3) 利用与 $T(\lambda)$ 对应的另一标量实有理函数 $q(x)$ 的关联分块矩阵 $C_q$ 的数值半径不等式，亦获得一个上界。当系数矩阵为酉矩阵时，对这些界进行了比较。文中给出了数值算例以说明所得结果。