= Instantaneous stretching rate
{title2=$\sigma(e)=e^T(J+J^T)e/2$}
For a unit perturbation direction $e$ and <Jacobian matrix> $J$, the instantaneous logarithmic growth rate of its Euclidean norm is $e^TSe$, where $S=(J+J^T)/2$. Indeed $d|\eta|^2/dt=2\eta^TS\eta$ for $\dot\eta=J\eta$. The <eigenvalues> of the <symmetric matrix> $S$ give the principal rates and its orthonormal <eigenvectors> give the principal directions. These directions generally rotate as the trajectory evolves; their instantaneous rates need not equal asymptotic <Lyapunov exponents>.
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