Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system: $$ \frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n, $$ where $\max\{\cdot\}$ acts element-wise on vectors.

True or false. If all possible diagonal blocks of $-I+A$ (including $-I+A$ itself and the diagonal entries of $-I+A$) have eigenvalues with (strictly) negative real part, then $x(t)\to 0$ as $t\to\infty$, for all $x(0)$.

A large amount of numerical simulations support the above claim. However a formal proof still remains elusive to me.


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