Bounds on the Moduli of Eigenvalues of Rational Matrices

A rational matrix is a matrix-valued function \(R(\lambda ): {\mathbb {C}} \rightarrow M_p\) $R(\lambda ):\mathbb{C}\to M_p$ such that \(R(\lambda ) = \begin{bmatrix} r_{ij}(\lambda ) \end{bmatrix} _{p\times p}\) $R(\lambda )={\left[\begin{array}{c}{r}_{\mathit{ij}}(\lambda )\end{array}\right]}_{p\times p}$ , where \(r_{ij}(\lambda )\) $r_{\mathit{ij}}(\lambda )$ are scalar complex rational functions in \(\lambda \) $\lambda$ for \(i,j=1,2,\ldots ,p\) $i,j=1,2,\dots ,p$ . 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 \(R(\lambda )\) $R(\lambda )$ we associate a block matrix \({\mathcal {C}}_R\) ${\mathcal{C}}_R$ whose blocks consist of the coefficient matrices of \(R(\lambda )\) $R(\lambda )$ , as well as a scalar real rational function q(x) whose coefficients consist of the norm of the coefficient matrices of \(R(\lambda )\) $R(\lambda )$ . We prove that a zero of q(x) which is greater than the moduli of all the poles of \(R(\lambda )\) $R(\lambda )$ will be an upper bound on the moduli of eigenvalues of \(R(\lambda )\) $R(\lambda )$ . 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 \(R(\lambda )\) $R(\lambda )$ .