Computing generalized inverses using LU factorization of matrix product

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

The mixed-type reverse order laws for generalized inverses of the product of two matrices

“…In [3,5,[24][25][26], the reverse order laws for {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}-inverses were considered. For other interesting results on this subject see [2,9,10,13,18]. In this paper, by applying the maximal ranks of generalized Schur complement [18,19], we obtain necessary and sufficient conditions for the following mixed-type reverse order laws:…”

The Forward Order Law for Least Squareg-Inverse of Multiple Matrix Products

“…The reverse order law for the generalized inverse of multiple matrix products yields a class of interesting problems that are fundamental in the theory of the generalized inverse of matrices; see [2,3,5,8,9]. As one of the core problems in reverse order laws, the necessary and sufficient conditions for the reverse order laws for the generalized inverse of matrix product hold, is useful in both theoretical study and practical scientific computing, this has attracted considerable attention, and many interesting results have been obtained; see [8,[10][11][12][13][14][15][16][17][18].…”

The forward order laws for {1,2,3}- and {1,2,4}-inverses of a three matrix products

“…We refer the reader to [1,13] for basic results on the generalized inverses. Theory and computations of the reverse order laws for generalized inverses of matrix product are important subjects in many branches of applied science, such as non-linear control theory, matrix theory, matrix algebra, see [6,8,9,13]. One of the core problems in reverse order laws is to find the necessary and sufficient conditions for the reverse order laws for the generalized inverse of matrix product and it has attracted considerable attention, see [1,2,7,9,18,19].…”

“…During the recent years, Zheng and Xiong [18,19] studied the reverse order laws for {1, 2, 3}-inverses and {1, 2, 4}-inverses of multiple products. For other interesting results on this subject see [1,2,8,9,14].…”