Symmetric-part contraction criterion ($\frac{1}{2}(A+A^T)⪯-\kappa I\Rightarrow \Vert e^{At}v\Vert \le e^{-\kappa t}\Vert v\Vert$)

For $\dot{e}=Ae$, differentiation gives $d\Vert e{\Vert }^2/dt=e^T(A+A^T)e$. A uniform negative-definiteness bound with $\kappa >0$ yields exponential contraction by direct integration of this differential inequality. This is a sufficient criterion for asymptotic stability, stronger than requiring all eigenvalues of $A$ to have negative real parts. A merely semidefinite symmetric part needs additional analysis.