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A new approach for solving electromagnetic wave propagation problems is currently being developed at the Naval Command, Control and Ocean Surveillance Center (NCCOSC), RDT and E Division (NRaD). This new approach is based upon an 8 by 8 matrix representation of the Maxwell field equations. In addition, a computer software package based on this matrix representation of electromagnetic theory is also being written and tested at NRaD to handle a variety of scenarios involving electromagnetic wave propagation through matter. This software package is referred to as the MATURE Program. MATURE is the acronym for Matrix Approach To Understanding Relativistic Electrodynamics. The MATURE Program is written in MATLAB code for use on a Sun 4 SPARCstation 2 workstation. Under Independent Research (IR) FY 92 funding, this matrix approach was successfully employed in solving problems dealing with electromagnetic wave propagation through dielectric, crystalline, linear electro-optic, and magneto- optic materials of infinite extent. Under the Office of Naval Research (ONR) FY 93 funding, this matrix formulation was extended to handle problems involving wave propagation through multilayer dielectric media with planar boundaries. Presented in this technical document is the underlying theory of this matrix approach. Several numerical examples, based on the use of the MATURE Program, are also included to illustrate the use of the matrix approach in solving electromagnetic wave propagation problems.