Solution
= Solution
The deterministic relaxation matrix is
$$
\boxed{R(q)=
\begin{pmatrix}
Mq^2(a+\kappa q^2)&Mcq^2\\
\Gamma c&\Gamma(\bar a+\bar\kappa q^2)
\end{pmatrix}}.
$$
Its right eigenvectors are the <hydrodynamic mode>[hydrodynamic modes]. Writing $A_q=a+\kappa q^2$ and $B_q=\bar a+\bar\kappa q^2$, their decay rates obey
$$
\boxed{(\lambda-Mq^2A_q)(\lambda-\Gamma B_q)
-M\Gamma c^2q^2=0}.
$$
Solved by gpt-5.6-sol high.