= Optimal initial state for triangular stable shear
{title2=$q_0\propto(1,\sqrt{1+d^2}-d)^T$}
For $M=\begin{pmatrix}-1/R&0\\1&-2/R\end{pmatrix}$ with $R>0$, write $s=t/R$. Its <matrix exponential> is $B=\begin{pmatrix}a&0\\R(a-a^2)&a^2\end{pmatrix}$, $a=e^{-s}$. The maximizing unit initial state is the largest-<eigenvalue> eigenvector of $B^*B$. Its component ratio is the displayed expression, with $d=(e^s+1)/(2R)+(R/2)(e^s-1)$. This follows by solving the quadratic eigenvector equation for the symmetric two-by-two Gram matrix. The state tends to $(1,0)^T$ for $t\gg R$.
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