Instantaneous stretching rate ($\sigma (e)=e^T(J+J^T)e/2$)

For a unit perturbation direction $e$ and Jacobian matrix $J$, the instantaneous logarithmic growth rate of its Euclidean norm is $e^{T}Se$, where $S=(J+J^T)/2$. Indeed $d|\eta {|}^2/dt=2{\eta }^{T}S\eta$ 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.