Asymptotic expressions for the reflected, transmitted, and internal scattered radiation field in optically thick, vertically homogeneous, plane-parallel media are derived from first principles by using the discrete ordinates method of radiative transfer. Compact matrix equations are derived for computing the escape function, diffusion pattern, diffusion exponent, and the reflection function of a semi-infinite atmosphere in terms of the matrices, eigenvectors, and eigenvalues that occur in the discrete ordinates method. These matrix equations are suitable for numerical computations and are valid throughout the full range of single scattering albedos. The present formulations are validated by comparing them with established methods of radiative transfer.