  The equations for the electromagnetic field in an anisotropic media are
written in a form containing only the transverse field components relative to a
half plane boundary. The operator corresponding to this formulation is the
electromagnetic system's matrix. A constructive proof of the existence of
directional wave-field decomposition with respect to the normal of the boundary
is presented.
  In the process of defining the wave-field decomposition (wave-splitting), the
resolvent set of the time-Laplace representation of the system's matrix is
analyzed. This set is shown to contain a strip around the imaginary axis. We
construct a splitting matrix as a Dunford-Taylor type integral over the
resolvent of the unbounded operator defined by the electromagnetic system's
matrix. The splitting matrix commutes with the system's matrix and the
decomposition is obtained via a generalized eigenvalue-eigenvector procedure.
The decomposition is expressed in terms of components of the splitting matrix.
The constructive solution to the question on the existence of a decomposition
also generates an impedance mapping solution to an algebraic Riccati operator
equation. This solution is the electromagnetic generalization in an anisotropic
media of a Dirichlet-to-Neumann map.
