INVERS TERGENERALISASI MOORE PENROSE

Abstract

The generalized inverse is a concept for determining the inverse of a singular matrix and and  matrix which has the characteristic of the inverse matrix. There are several types of generalized inverse, one of which is the Moore-Penrose inverse. The matrix  is called Moore Penrose inverse of a matrix if it satisfies the four penrose equations and is denoted by . Furthermore, if the matrix  satisfies only the first two equations of the Moore-Penrose inverse and , then  is called the group inverse of  and is denoted by . The purpose of this research was to determine the group inverse of a non-diagonalizable square matrix using Jordan’s canonical form and Moore Penrose’s inverse of a singular matrix, also a non-square matrix using the Singular Value Decomposition (SVD) method. The results of this study are the sufficient condition for a matrix  to have a group inverse, i.e., a matrix  has an index of 1 if and only if the product of two matrices forming  is a full rank factorization and is invertible. Whereas for a singular matrix  and a non-square , the Moore-Penrose inverse can be determined using Singular Value Decomposition (SVD).

Keywords: generalized matrix inverse, Moore Penrose inverse, group inverse, Jordan canonical form, Singular Value Decomposition.