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Many scientific problems, including those in physics, astrophysics, chemistry, and information processing, involve the solutions of eigen-systems and the determination of sensitivity of those eigen-systems. The latter requires tangent-linear and adjoint formulations of eigen-systems. There are several numerical packages, such as Numerical Recipes and LAPACK, available for calculating eigenvalues and eigenvectors in the forward model. This paper presents analytical expressions for the tangent-linear and adjoint eigenvalue, and eigenvector in the radiative transfer models. The algorithm is called “advanced matrix operator method (AMOM)” and has been implemented to replace the double-adding method for calculating layer source functions, transmittance, as well as reflection matrices in the Community Radiative Transfer Model (CRTM) developed at the Joint Center for Satellite Data Assimilation, United States. The CRTM has been operationally used for satellite radiance assimilation at the National Oceanic and Atmospheric Administration (NOAA) in supporting daily weather forecasting and satellite products. This CRTM is also widely used in radiative transfer community around the world. Using the analytic approach in the CRTM tangent-linear and adjoint calculations returns the same result as the advanced doubling-adding method from the previous CRTM version, but this new approach reduces computation time by 23%-134% in cloudy radiance calculations. With the AMOM, the CRTM is now applicable for visible sensors.