= Optimal energy amplification of a linear system
{title2=$G(t)$}
For the <matrix exponential> $A(t)=e^{Lt}$, maximizing the energy ratio over nonzero initial conditions gives
$$
G(t)=\max_{x_0\ne0}\frac{x_0^*A(t)^*A(t)x_0}{x_0^*x_0}=\|A(t)\|_2^2=\sigma_{\max}(A(t))^2.
$$
The <Rayleigh quotient> achieves its maximum at an <eigenvector> of $A^*A$ with largest <eigenvalue>, equivalently a <right singular vector> of $A$. Such an optimal disturbance is generally not an <eigenvector> of $L$.
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