Let's consider the following system:
$${\boldsymbol x}(t)=e^{-\boldsymbol A t}{\boldsymbol x_0}$$
Where ${\boldsymbol A}$ is a square matrix (potentially asymmetric).
What are the required and sufficient conditions on ${\boldsymbol A}$ for having $||{\boldsymbol x}(t)||$ always shrinking with time monotonically.
I suspect the required and sufficient condition is that ${\boldsymbol A}$ is positive definite. But I cannot prove it.
PS.
I cannot conclude if the system is stable, the state monotonically shrinks.
If we let $\phi(t) = {1 \over 2} \|x(t)\|^2$, then $\phi'(t) = x^T(t) x'(t) = -x^T(t) A x(t)$.
Then, the norm of the state is strictly decreasing for any non zero initial state $x_0$ iff $-x_0^T A x_0 < 0$ for all $x_0 \neq 0$.
Note that $x_0^T A x_0 > 0$ for all $x_0 \neq 0$ iff $x_0^T (A+A^T) x_0 > 0$ for all $x_0 \neq 0$ iff $A>0$.