= Normality criterion for a two-mode shear model
{title2=$a^2=c^2,\quad b(a+c)=0$}
For $A=\begin{pmatrix}-i\omega_1&a\\c&-i(\omega_1-b)\end{pmatrix}$ with real parameters,
$$
AA^\dagger-A^\dagger A=\begin{pmatrix}a^2-c^2&-ib(a+c)\\ib(a+c)&c^2-a^2\end{pmatrix}.
$$
Thus the displayed conditions are necessary and sufficient for a <normal matrix>. In the family $a=1$, $b=c=q$, normality requires $q=-1$, not $q=1$. If $q=10/Re$, this is the formal value $Re=-10$ and no positive <Reynolds number> qualifies. At that value $A$ is a <skew-Hermitian matrix> and its exponential is unitary, so every initial direction has <optimal energy amplification of a linear system> equal to one.
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