 Optimal initial state for triangular stable shear

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 $t\gg R$.