Bounds on the eigenvalues of matrix rational functions

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.