The Sensitivity of the Matrix Exponential

Abstract

In this paper we examine how the matrix exponential $e^{At} $ is affected by perturbations in A. Elementary techniques using $\log $ norms and the Jordan and Schur factorizations indicate that $e^{At} $ is least sensitive when A is normal. Through the formulation of an exponential condition number, more insight is gained into the complicated connection between the condition of the eigensystem of A and the sensitivity of $e^{At} $.