  Most nonlinear eigenvalue problems are rational eigenvalue problems or can be
approximated to rational eigenvalue problems. Bounds on the eigenvalues of
matrix rational functions (also known as rational matrices) are less studied in
literature compared to matrix polynomials. Our aim is to derive bounds on the
eigenvalues of an arbitrary matrix rational function in terms of its poles. A
bound is obtained as a root of a real rational function associated with a
matrix rational function. In addition, a block matrix associated with this
rational function is used to get explicit bounds on its zeros and in turn on
the eigenvalues of matrix rational function. Bounds on the zeros of an
arbitrary complex rational function are also obtained in this process.

Spectral Theory