Conditions for Existence, Representations, and Computation of Matrix Generalized Inverses

Abstract

Conditions for the existence and representations of {2}-, {1}-, and {1, 2}-inverses which satisfy certain conditions on ranges and/or null spaces are introduced. These representations are applicable to complex matrices and involve solutions of certain matrix equations. Algorithms arising from the introduced representations are developed. Particularly, these algorithms can be used to compute the Moore-Penrose inverse, the Drazin inverse, and the usual matrix inverse. The implementation of introduced algorithms is defined on the set of real matrices and it is based on the Simulink implementation of GNN models for solving the involved matrix equations. In this way, we develop computational procedures which generate various classes of inner and outer generalized inverses on the basis of resolving certain matrix equations. As a consequence, some new relationships between the problem of solving matrix equations and the problem of numerical computation of generalized inverses are established. Theoretical results are applicable to complex matrices and the developed algorithms are applicable to both the time-varying and time-invariant real matrices.

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme

“…To find the unknown weighting coefficients in (15) and, more specifically, in (16), we need to solve a symbolic problem. As such, a Mathematica code [28] was employed to do such a task, as follows: This was given only to ease understanding of the procedure of obtaining the coefficient.…”

“…Before providing the main results concerning the convergence analysis of the proposed scheme, we furnish the following lemmas, inspired by [29], which reveal how the iterates generated by (15) have some specific important relations and, then, show a relation between (4) and (15).…”

“…Subsequently, now the relation is valid for k > 0, then we discuss that it will still be true for k + 1. Taking our matrix iteration (15) into consideration, we have:…”

“…The objective of this section is to provide a matrix analysis for the convergence of the iteration scheme (15). Theorem 1.…”

“…The sequence produced by (34) is the result of employing (15) in calculating the zero σ −1 i of the function…”