Is anything known about these types of linear operators?

Question

Let $A$ be an $n\times n$ matrix, and let $x_{0}\in\mathbb{R}^{n}$, for some $n$. Consider $$\frac{dx}{dt}=Ax,\,x(0)=x_{0}$$ with solution $x=e^{At}x_{0}$. If there is a norm $\Vert\cdot\Vert$ such that $\Vert e^{At}x\Vert<\Vert x\Vert$ for all $t>0$ and all $x\ne0$, then the solution $x(t)$ approaches zero with decreasing norm. Is anything known about this type of matrix $A$?