Bounds on the Moduli of Eigenvalues of Rational Matrices

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 \(R(\lambda )\).