Let $A(x)$ be a real valued matrix with $x$-dependent coefficients where $x\in \mathbb{R}$. What is the necessary and sufficient condition on $A(x)$ such that the matrix exponential
$$\exp( - A(x))$$
is bounded?

I think it should be that there is some constant matrix $B$ so that $A(x) +B$ is a positive matrix or is positive definite. This comes down to what is going on in the non-diagonal terms and I can't see how to deal with it.

Thanks :)