= Symmetric-part contraction criterion
{title2=$\tfrac12(A+A^T)\preceq-\kappa I\ \Rightarrow\ \|e^{At}v\|\leq e^{-\kappa t}\|v\|$}
For $\dot e=Ae$, differentiation gives $d\|e\|^2/dt=e^T(A+A^T)e$. 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.
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