A constrained conjugate gradient method and the solution of linear equations

A conjugate gradient method for solving minimization problems subject to linear equality constraints is developed. The method can be considered as an extension of the unconstrained Fletcher and Reeves algorithm. Proofs of several theorems related to the method are given. An application of the method is to provide an alternative method to solve a real system of linear equations for varying right-hand sides. Numerical solutions can be obtained in one iteration which leads to the development of a new direct method to invert a square matrix. Since this direct method involves matrix-vector multiplications and outer products which are easy to parallelize, it has an advantage over existing well-known direct methods. Numerical examples are given.