Relationship negative definite matrix and negative mass balance

Question

i'm currently working with linear differential equations of the the form
$\frac{dA}{dt}=MA$,
where $M\in\mathbb{R}^{n\times n}$.
The solution of this equation is also quit simple
$A(t)=e^{Mt}A(0)$
with $A(t)=(A_1(t),...,A_n(t))^T$.
Intuitivly the system losses mass, i.e. $\sum_i A_i(t)$ is strictly monoton decreasing for all $ A(0)\in \mathbb{R}_{+,0}^m/{0}$, if and only if M is negative definite. The direction negative definite => System losses mass follows from the negative eigenvectors. But i can't think of a proof for the other direction. Or is the other direction not true in general?
Thanks:)