The Moore-Penrose Inverse Matrix of the Non-Normal Cyclic Subgroup Graph by Using Rank Factorization

Abstract

The inverse matrix can be determined for any non-singular or square matrix and it is particularly useful in solving linear systems. At the same time, the Moore-Penrose inverse matrix is an inverse matrix for any rectangular matrix constructed when the linear system has more unknowns than the number of values provided by the measurements. In previous studies, this inverse matrix has been applied to some graphs unrelated to finite groups. The main objective of this paper is to determine the Moore-Penrose inverse matrix in general form for the non-normal cyclic subgroup graph of two finite groups, namely the dihedral groups and the generalized quaternion groups. The digraph for these two finite groups represented by the incidence matrix is then applied to generate the Moore-Penrose inverse matrix using the rank factorization method. It is useful when the matrix is not of full rank. The Moore-Penrose inverse matrices are given in the form of matrix size, and the entries of the matrices are block matrices, as the digraph is disconnected and the incidence matrix is not unique.