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 Rouché’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.
