= Two-dimensional triangular model of transient growth
For $L=\begin{pmatrix}\lambda_1&0\\1&\lambda_2\end{pmatrix}$ with distinct real negative <eigenvalues>, the <matrix exponential> is $A=\begin{pmatrix}a&0\\b&d\end{pmatrix}$, where $a=e^{\lambda_1t}$, $d=e^{\lambda_2t}$ and $b=(a-d)/(\lambda_1-\lambda_2)$. The <optimal energy amplification of a linear system> is
$$
G(t)=\frac{a^2+b^2+d^2+\sqrt{(a^2+b^2+d^2)^2-4a^2d^2}}2.
$$
Its <eigenvectors> are <nonorthogonal>, and immediate energy growth occurs exactly when $\lambda_1\lambda_2<1/4$. If $\lambda_2>\lambda_1$, putting $\alpha=(\lambda_2-\lambda_1)^{-1}$ gives $G(t)\sim(1+\alpha^2)e^{2\lambda_2t}$. The optimal asymptotic initial direction $(\alpha,1)$ is an <adjoint eigenvector>, while the eventual state aligns with the right <eigenvector> $(0,1)$.
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