Cholesky Factorization and Matrix Inversion


Book Description

The Cholesky square root algorithm used in the solution of linear equations with a positive definite matrix of coefficients is developed by elementary matrix algebra, independent of the Gaussian elimination from which it was originally derived. The Cholesky factorization leads to a simple inversion procedure for the given matrix. A simple transformation makes the inversion applicable to nonsymmetric matrices. The least squares hypothesis is shown to be the simplest and most general unique solution of a system of linear equations with a nonsquare matrix of coefficients. The method of proof is extended to develop the Gaussian elimination algorithm in a readily comprehensible procedure.







Radiation Pressure on a Spheroidal Satellite


Book Description

The pressure of solar radiation on a spherical balloon satellite is proportional to its cross-sectional area. However, there is evidence to indicate that the PAGEOS satellite has not remained spherical, but is more nearly a prolate spheriod that is rotating about its minor axis. If this is true, the force of solar radiation incident upon its surface must be expressed in terms of the shape of the surface and its orientation to the sun. Furthermore, radiation reflected from the surface of an aspherical balloon will impart an additional force which can be large enough to significantly perturb the orbit. By starting with basic equations for the radiation forces on a flat plate and integrating over the sunlit portion of the surface, exact expressions are obtained for both the incident and reflected radiation forces on a stationary, prolate spheroidal satellite. These expressions are then used to develop power series expansions for the radiation forces on a rotating spheroid.