Computing generalized inverses using LU factorization of matrix product

TL;DR

This paper introduces an algorithm for computing various generalized inverses, including the Moore-Penrose inverse, of rational matrices using LU and Cholesky factorizations, with implementation and comparison to existing methods.

Contribution

It presents a novel algorithm based on matrix product representations and Cholesky factorization for computing generalized inverses of rational matrices.

Findings

Algorithm successfully computes generalized inverses.

Numerical results are comparable or superior to existing methods.

Implementation in MATHEMATICA and DELPHI demonstrates practicality.

Abstract

An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the algorithm, corresponding to the Moore-Penrose inverse, are compared with corresponding results obtained by several known methods for computing the Moore-Penrose inverse.