I'm studying waves moving through mediums and I come across this type of equation a lot while describing acoustic waves speed and stress on the material,
$$\begin{gather} \frac{\partial}{\partial z}\begin{bmatrix} A(t,z) \\ B(t,z) \end{bmatrix} = -\frac{1}{c(z)} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} A(t,z) \\ B(t,z) \end{bmatrix} \end{gather}$$
and the trace of the matrix on the right hand side, always equals $0.$ Is there a reason for this which is related to energy conservation or something similar? $A$ and $B$ are left and right going waves and $c(z)$ is the wave speed.
I am working on a problem of my own and trying to follow the same theory for elastic waves in mediums, but when I get a similar equation, the matrix on the right hand side does not have a trace of $0$.
Thanks in advance.