Generalized inverse and it’s applications to the solutions of system of linear equations and semigroup

Abstract

There are a number of versatile generalizations of the usual inverse matrix, referred to in this thesis as generalized inverse matrices. The definitions and properties of some of the common generalized inverse matrices are described, including methods for constructing them. A number of applications are discussed, including their use in solving consistent systems of linear equations which do not have the same number of equations as variables, or which have a singular coefficient determinant. A certain type of generalized inverse is shown to give the least‐squares solution of an inconsistent system of linear equations. Other applications are to systems of nonlinear equations, to integer solutions of systems of equations and to linear programming. The purpose of the thesis is to show that a singular or 𝑚×𝑛 matrix has a generalized inverse (g-inverse). A matrix 𝐴− is said to be a generalized inverse if it fulfils the condition 𝐴𝐴−𝐴=𝐴. The raw canonical system is used to find the generalized inverse. So we will be using theorems, examples and programming language to prove and solve G-inverse. This thesis will also consist of semigroups, contour integration and applications which is related generalized inverse.