Eigenvalue Conditions for Convergence of Singularly Perturbed Matrix Exponential Functions

Abstract. We investigate convergence of sequences of n × n matrix exponential functions t → etA k for t > 0, where Ak → A, Ak is nonsingular and A is nilpotent. Specifically, we address pointwise convergence, almost uniform convergence, and, viewing the exponential as a Schwartz distribution, weak∗ convergence. We show that simple results can be obtained in terms of the eigenvalues of A−1 k alone. In particular, a necessary and sufficient condition for weak ∗ convergence in terms of eigenvalue behavior is attainable. We then apply our results to real-analytic matrices A(ε) as ε → 0+. Our work is applicable to matrices over both R and C.