TITLE: What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$?
QUESTION [6 upvotes]: What property of a matrix causes $\|e^{tA}\|_2$ to oscillate as $t\rightarrow\infty$?
The best I can come up with is that $A=bi\cdot M$ for $b$ a non-zero real number and $M$ a non-zero idempotent matrix, since in that case we have:
$$\|e^{tA}\|_2 =  \left\|\sum_{i=0}^{\infty}\frac{(tbi)^k\cdot M^k}{k!}\right\|_2= \left\|I+M\sum_{i=1}^{\infty}\frac{(tbi)^k}{k!}\right\|_2= \|I+Me^{tbi}\|_2,$$
which quite clearly oscillates.  However it's certainly not clear to me that this is the only way to obtain oscillation, and if it is, I don't see how to go about proving it.

REPLY [2 votes]: Most non-normal matrices whose eigenvalues are pure imaginary will have this property.  (Normal means $AA^* = A^*A$.)
E.g. $A = P D P^{-1}$ where $P$ is an invertible $2\times 2$ matrix that is not a multiple of a unitary matrix, and $D = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$: then
$$ e^{tA} = P \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} P^{-1} = \cos(\theta)I + \sin(\theta) B ,$$
with $B = P D P^{-1}$.
E.g. $A = \begin{bmatrix} 0 & \lambda-1 \\ \lambda+1 & 0 \end{bmatrix}$ if $-1 < \lambda < 1$.