A generalized iterative method to compute the inverse of an invertible matrix

Abstract

The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, and even for designing computer game graphics. Matrices are very useful due to the fact that they can be easily manipulated. We use the notation A-1 to denote the inverse of a matrix A. One of the major uses of inverses is to solve a system of linear equations. You can write a system in matrix form as AX=B, then X=A-1B. Inverses are also used in communication through coded messages. The use of coding has become particularly significant in recent years (due to the explosion of internet for example). One way to code a message is to use matrices and their inverses. Indeed, consider a fixed invertible matrix A. Convert the message into a matrix B such that AB is possible to perform. Send the message generated by AB. At the recipient end, they will need to know A-1 in order to decode the message sent. Indeed, we have A-1 (AB) =B, which is the original message. There are two classes of methods for finding the inverse matrices. Direct methods; a finite number of arithmetic operations leads to an exact solution. Examples of such direct methods include Gauss elimination, Gauss-Jordan elimination, the matrix inverse method and LU factorization. Methods of the second type are called Iterative methods. Iterative methods start with an arbitrary first approximation to the unknown solution. These methods are used for finding the inverse matrices of large systems of equations. In this thesis, we will work on Matrix, Matrix norm, Norms and Matrix Inversion. We did some examples on how to find the inverse of a matrix using a direct method. We recalled the definition of order of convergence throughout an example. Then in chapter 2, we showed that the following iteration method Xn+1=∑_(i=0)^p▒Xn (I -AXn )i . Converges to A-1 under the assumption ||I - AX0|| < 1. We also showed that the order of convergence is p+1. Then we found an upper bound on the norm of difference between m's iteration and the exact value of the Matrix inverse. This generalizes the results presented in [15], where the author only considered the case p = 1. First, we dealt with case p = 2, then the case p = 3. We also illustrate with a simple numerical analysis how the iteration works. Afterwards, we generalize the results presented in [15] for arbitrary p.