  A rational matrix is a matrix-valued function $R(\lambda): \mathbb{C}
\rightarrow M_p$ such that $R(\lambda) = \begin{bmatrix} r_{ij}(\lambda)
\end{bmatrix}_{p\times p}$, where $r_{ij}(\lambda)$ are scalar complex rational
functions in $\lambda$ for $i,j=1,2,\ldots,p$. The aim of this paper is to
obtain bounds on the moduli of eigenvalues of rational matrices in terms of the
moduli of their poles. To a given rational matrix $R(\lambda)$ we associate a
block matrix $\mathcal{C}_R$ whose blocks consist of the coefficient matrices
of $R(\lambda)$, as well as a scalar real rational function $q(x)$ whose
coefficients consist of the norm of the coefficient matrices of $R(\lambda)$.
We prove that a zero of $q(x)$ which is greater than the moduli of all the
poles of $R(\lambda)$ will be an upper bound on the moduli of eigenvalues of
$R(\lambda)$. Moreover, by using a block matrix associated with $q(x)$, we
establish bounds on the zeros of $q(x)$, which in turn yields bounds on the
moduli of eigenvalues of $R(\lambda)$.
