Optimal initial state for triangular stable shear ($q_0\propto (1,\sqrt{1+d^2}-d{)}^T$)

For $M=\left(\begin{array}{cc}-1/R& 0\\ 1& -2/R\end{array}\right)$ with $R>0$, write $s=t/R$. Its matrix exponential is $B=\left(\begin{array}{cc}a& 0\\ R(a-a^2)& a^2\end{array}\right)$, $a=e^{-s}$. The maximizing unit initial state is the largest-eigenvalue eigenvector of $B^{\ast }B$. 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 $t\gg R$.

In the optimal initial state for triangular stable shear, expansion of $e^{t/R}$ gives the displayed leading parameter with relative error $O(t/R)$. Keeping both terms distinguishes a large-$R$ shear regime from the limit $t\to 0$ at fixed $R$. The latter direction is the largest-eigenvalue direction of the Hermitian part of $M$; the former can instead have large absolute time and substantial shear amplification even though $t/R$ is small.