The matrices highlighted in Yellow are the matrices I don't understand how to find.
What I don't understand is how the matrices in the picture are found. The equations below are the the equations I am trying to get into the form like the picture.
$$ \frac{\partial U}{\partial t} + \frac{\partial F_c}{\partial x} + \frac{\partial G_c}{\partial y} = S_v $$
Where $$ U = \begin{pmatrix} \rho \\ \rho u \\ \rho v \\ \rho E \end{pmatrix}, F_c = \begin{pmatrix} \rho u \\ \rho u^2 + p \\ \rho vu \\ (\rho E + p)u \end{pmatrix}, G_c = \begin{pmatrix} \rho v \\ \rho uv \\ \rho v^2 + p \\ (\rho E + p)v \end{pmatrix} $$
We can ignore the $S_v$ term. What I don't understand is how to get what is in the picture. I hope this is enough to help me.
The characteristic form is $\frac{\partial Q}{\partial t}$ + F$\frac{\partial Q}{\partial x}$ + G$\frac{\partial Q}{\partial y}$ = $S_v$
where $$ F = \begin{pmatrix} u&{\rho}&0&0 \\ 0&u&0&1/{\rho} \\ 0&0&u&0 \\ 0&{\rho}c^2&0&u \end{pmatrix}, G = \begin{pmatrix} v&0&{\rho}&0 \\ 0&v&0&0 \\ 0&0&v&1/{\rho} \\ 0&0&{\rho}c^2&v \end{pmatrix} $$