What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a square matrix?

Question

I was doing a matrix calculation and need to find

$$\lim_{t\to \infty} \mathrm{e}^{At}=?$$

What is the limit of $\mathrm{e}^{At}$ when $t\to \infty$, for $A$ a matrix?

Answer 1

The limit is $0$ if and only if $A$ is Hurwitz (ie. real part of any eigenvalue of $A$ is negative). To see why this is true, apply the Spectral Mapping Theorem, which basically says the eigenvalues of $e^{At}$ is of the form $e^{\lambda t}$ where $\lambda$ is an eigenvalue of $A$.

Answer 2

This limit depends on the eigenvalues of $A$.

Let $A$ be diagonalizable, i.e. $A=U^{-1}DU$, $D=\mathrm{diag}(d_1,\ldots,d_n)$. Then $$ \mathrm{e}^{tA}= U^{-1}\mathrm{e}^{tD}U=U^{-1}\mathrm{diag}(\mathrm{e}^{d_1t},\ldots,\mathrm{e}^{d_nt})U. $$ So, the limit of $\mathrm{e}^{tA}$ depends on the $d_1,\ldots,d_n$. If all their real parts are negative, then $\mathrm{e}^{tA}\to 0$.