In a paper that I'm reading, the author mentions something of this sort:-
...we arrive at an eigenvalue problem defined by the following matrix equation:
$ \left[ {\begin{array}{ccc} \sinh(\beta) & \sin(\gamma) \\ \beta \psi_\beta \sinh(\beta) & \gamma \psi_\gamma \sin(\gamma) \\ \end{array} } \right] \left[ {\begin{array}{cc} C_2\\ C_4\\ \end{array} } \right] =0 $
The eigenvalues are obtained by setting the determinant of the matrix to $0$ and then solving the characteristic equation. By solving the characteristic equation, one obtains $\sin(\gamma) = 0 \implies \gamma =n\pi $.
I admit I haven't had an introductory course in Linear Algebra yet, but I believe eigenvalues are $\lambda$ satisfying the matrix equation $Ax=\lambda x$, for some matrix $A$. But in the above case, the matrix equation is in the form $Ax=0$. Besides, obtaining $\sin(\gamma)=0$ is as simple as setting the determinant = $0$ (assuming that non-trivial solutions exist). How do eigenvalues and the characteristic equation come into the picture here?