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Complexity
Volume 2017, Article ID 6429725, 27 pages
https://doi.org/10.1155/2017/6429725
Research Article

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

1Faculty of Science and Mathematics, Department of Computer Science, University of Niš, Višegradska 33, 18000 Niš, Serbia
2Faculty of Computer Science, Goce Delčev University, Goce Delčev 89, 2000 Štip, Macedonia
3Aristoteleion Panepistimion, Thessalonikis, Greece

Correspondence should be addressed to Predrag S. Stanimirović; sr.ca.in.fmp@okcep

Received 3 January 2017; Accepted 18 April 2017; Published 5 June 2017

Academic Editor: Sigurdur F. Hafstein

Copyright © 2017 Predrag S. Stanimirović et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Conditions for the existence and representations of -, -, and -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.