Applications of the Moore-Penrose Generalized Inverse to Linear Systems of Algebraic Equations

Abstract

In this work, we consider linear systems of algebraic equations. These systems are studied utilizing the theory of the Moore-Penrose generalized inverse or shortly (MPGI) of matrices. Some important algorithms and theorems for computation the MPGI of matrices are given. The singular value decomposition (SVD) of a matrix has a very important role in computation the MPGI, hence it is useful to study the solutions of over- and under-determined linear systems. We use the MPGI of matrices to solve linear systems of algebraic equations when the coefficients matrix is singular or rectangular. The relationship between the MPGI and the minimal least squares solutions to the linear system is expressed by theorem. The solution of the linear system using the MPGI is often an approximate unique solution, but for some cases we can get an exact unique solution. We treat the linear algebraic system as an algebraic equation with coefficients matrix A (square or rectangular) with complex entries. A closed form for solution of linear system of algebraic equations is given when the coefficients matrix is of full rank or is not of full rank, singular square matrix or non-square matrix. The results are taken from the works mentioned in the references. A few examples including linear systems with coefficients matrix of full rank and not of full rank are provided to show our studding.