We consider analytic solutions to Maxwell's equations in general homogeneous linear media. In general, the dielectric tenser is not Hermitian, and it is a function of the wavevector. We present our formalism for obtaining closed form analytic solutions for the eigenmodes in the general case. The dispersion relation is described by a quartic equation using covariant representation. Explicit expressions are obtained for the eigenmodes in terms of the propagation direction, the permittivity and the permeability. The results may be useful to describe the optical fields as well as energy and momentum propagation in a wide variety of optical elements, such as magneto-optical films, dye doped liquid crystals, gyrotropic, dichroic and magneto-chiral materials. A simple example of a birefringent dichroic crystal is considered.