Eigenvalue Conditions for Convergence of Singularly Perturbed Matrix Exponential Functions

Abstract

We investigate convergence of sequences of $n\times n$ matrix exponential functions $t\rightarrow e^{tA_{k}^{-1}}$ for $t>0$, where $A_{k}\rightarrow A$, $A_{k}$ is nonsingular and A is nilpotent. Specifically, we address pointwise convergence, almost uniform convergence, and, viewing the exponential as a Schwartz distribution, weak$^{\ast}$ convergence. We show that simple results can be obtained in terms of the eigenvalues of $A_{k}^{-1}$ alone. In particular, a necessary and sufficient condition for weak$^{\ast}$ convergence in terms of eigenvalue behavior is attainable. We then apply our results to real-analytic matrices $A(\varepsilon)$ as $\varepsilon\rightarrow0^{+}$. Our work is applicable to matrices over both $\mathbb{R}$ and $\mathbb{C}$.