Upper and lower bounds on absolute values of the eigenvalues of a matrix polynomial are well studied in the literature. As a continuation of this we derive, in this manuscript, bounds on absolute values of the eigenvalues of matrix rational functions using the following techniques/methods: the Bauer-Fike theorem, a Rouch$\text{\'e}$ theorem for matrix-valued functions and by associating a real rational function to the matrix rational function. Bounds are also obtained by converting the matrix rational function to a matrix polynomial. Comparison of these bounds when the coefficients are unitary matrices are brought out. Numerical calculations on a known problem are also verified.

翻译：矩阵多项式特征值绝对值的上下界在文献中已有充分研究。作为这一研究的延续，本文利用以下技术/方法推导了矩阵有理函数特征值绝对值的界：Bauer-Fike定理、矩阵值函数的Rouché定理，以及将实有理函数与矩阵有理函数相关联的方法。通过将矩阵有理函数转化为矩阵多项式，也得到了相应的界。本文还比较了当系数为酉矩阵时这些界的差异，并对已知问题进行了数值计算验证。