A Systematic Analysis on Analyticity of Semisimple Eigenvalues of Matrix-Valued Functions

Abstract

In this paper we study the existence of analytic eigenvalue functions of an analytic matrix-valued function $L(\lambda,\rho)$. Instead of proposing sufficient conditions for each individual case as in the literature, we propose a systematic scheme to discuss the existence of analytic eigenvalue functions of $L(\lambda,\rho)$ when $\lambda_0$ is a semisimple eigenvalue of $L(\lambda,0)$. We show that $\lambda(\rho)=\lambda_0+\rho\mu(\rho)$ is an eigenvalue of $L(\lambda,\rho)$ if and only if $\mu(\rho)$ is an eigenvalue of another analytic matrix-valued function $P(\mu,\rho)$ which is constructed based on the first order (partial) derivatives of $L(\lambda,\rho)$ at $(\lambda_0,0)$. Based on this result, a systematic scheme is proposed to check whether there exist analytic eigenvalue functions of $L(\lambda,\rho)$. This systematic scheme covers existing sufficient conditions in the literature, and can lead to much more general conditions.