Geometric Algorithm for Generalized Inverse of Rank Deficient Real Matrices

An inverse of rank deficient matrices (Generalized Inverse) is applied to solve ill-conditioned problems such as large-sized matrix computations. Generalized inverses have many applications in engineering problems, such as data analysis, electrical networks, character recognition, and so on. The most frequently used Moore-Penrose inverse matrices allow for solving such systems, even with rank deficiency, and they provide minimum-norm vectors as solutions. In this paper, we propose novel geometric algorithm for computing generalized inverse of rank deficient real matrices. While some of the approaches for the formulations are purely based on LU-factorization, the other variations are based on LU and QR factorizations. The uniqueness of the generalized inverse is also proved for the proposed formulations.