= Optimal time and gain of a Reynolds-scaled triangular model
{title2=$T^*\sim(4\log4/3)R,\quad G_{\max}\sim4^{-2/3}R^2$}
For $L=\begin{pmatrix}-1/R&1\\0&-1/(4R)\end{pmatrix}$ with $R>1$, put $s=\sqrt{1-R^{-2}}$ and $q=(5-3s)/(5+3s)$. The <optimal energy amplification of a linear system> reaches its maximum at
$$
T^*=-\frac{4R}{3}\log q,\qquad G_{\max}=\frac{1+s}{1-s}q^{5/3}.
$$
The maximizing initial orientation is $\rho^+$ and the final orientation is $\rho^-$ from the <orientation interval for transient energy growth>. Taking $R\to\infty$ gives the displayed viscous-time and Reynolds-squared asymptotics. These constants belong to this specified triangular model, not to every <non-normal matrix>.
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