Fast computing of the Moore-Penrose inverse matrix

Abstract

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m ×n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied. is defined. In the case when T is a real m × n matrix, Penrose showed that there is a unique matrix satisfying the four Penrose equations, called the generalized inverse of T. A lot of work concerning generalized inverses has been carried out, in finite and infinite dimension (e.g., (2, 11)). In this article, we provide a method for the fast computation of the generalized inverse of full rank matrices and of square matrices with at least one zero row or column. In order to reach our goal, we use a special type of tensor product of two vectors, that is usually used in infinite dimensional Hilbert spaces. Using this type of tensor product, we also give sufficient conditions for products of square matrices so that the reverse order law for the Moore-Penrose inverse ((1, 4, 5)) is satisfied. There are several methods for computing the Moore-Penrose inverse matrix (cf. (2)). One of the most commonly used methods is the Singular Value Decomposition (SVD) method. This method is very accurate but also time-intensive since it requires a large amount of computational resources, especially in the case of large matrices. In the recent work of P. Courrieu (3), an algorithm for fast computation of Moore- Penrose inverse matrices is presented based on a known reverse order law (eq. 3.2