I am given an anti-Hermitian and tridiagonal matrix $\bf M$ and I need to solve for a vector $\bf v$ such that
$$ \left[ \exp( t {\bf M} ) \, {\bf v} \right]_{1} = {\bf 0} $$
or at least $\approx {\bf 0}$. I am equally interested in the case where this holds $\forall t\ge0$ as well as after some finite $t_c$.
As of now, I do not have any clue how to schematically tackle this problem to get non-trivial solutions, especially for the second case.