Representations and geometrical properties of generalized inverses over fields

In this paper, as a generalization of Urquhart’s formulas, we present a full description of the sets of inner inverses and (B, C)-inverses over an arbitrary field. In addition, identifying the matrix vector space with an affine space, we analyze geometrical properties of the main generalized inverse sets. We prove that the set of inner inverses, and the set of (B, C)-inverses, form affine subspaces and we study their dimensions. Furthermore, under some hypotheses, we prove that the set of outer inverses is not an affine subspace but it is an affine algebraic variety. We also provide lower and upper bounds for the dimension of the outer inverse set.Agencia Estatal de InvestigaciónUniversidad de Alcal

Representations and symbolic computation of generalized inverses over fields

This paper investigates representations of outer matrix inverses with prescribed range and/or none space in terms of inner inverses. Further, required inner inverses are computed as solutions of appropriate linear matrix equations (LME). In this way, algorithms for computing outer inverses are derived using solutions of appropriately defined LME. Using symbolic solutions to these matrix equations it is possible to derive corresponding algorithms in appropriate computer algebra systems. In addition, we give sufficient conditions to ensure the proper specialization of the presented representations. As a consequence, we derive algorithms to deal with outer inverses with prescribed range and/or none space and with meromorphic functional entries.Agencia Estatal de investigaciónUniversidad de Alcal

Recent Theories and Applications in Approximation Theory

Soleymani, F.; Stanimirovic, PS.; Torregrosa Sánchez, JR.; Nik, HS.; Tohidi, E. (2015). Recent Theories and Applications in Approximation Theory. Scientific World Journal. (598279). doi:10.1155/2015/598279S59827

Weighted inner inverse for rectangular matrices

To extend the notation of inner inverses, we dene weighted inner in-verses of a rectangular matrix. In particular, we introduce aW-weighted (B;C)-inner inverse of A, for given matrices A;W;B;C, and present some characterizations andconditions for its existence. Since this new inverse is not unique, we describe the set of all W-weighted (B;C)-inner inverses of a given matrix. Several invertible matrix expressions which involve a  W-weighted (B;C)-inner inverse of A are studied. Us- ing such expressions, we represent a W-weighted (B;C)-inner inverse of some other matrix E. As a particular case of  W-weighted (B;C)-inner inverse, we investigate (B;C)-inner inverses of a rectangular matrix. We also establish some reverse order law properties  considering weighted inner inverses

Composite outer inverses for rectangular matrices

Various compositions of the Drazin inverse, the group inverse or the core-EP inverse with the Moore-Penrose inverse have investigated  last years. Solving some type of matrix equations, we introduce three new generalized inverses of a rectangular matrix, which are called the OMP, MPO and MPOMP inverses, because the outer inverse and the Moore-Penrose inverse are incorporated in their denition. As aconsequence, the notion of DMP, MPD, CMP and MPCEP inverses for a square matrix are covered by one general denition and extended to a rectangular matrix. We propose a common term, composite outer inverses, to denote such compositions of outer inverses and the Moore-Penrose inverse. Characterizations of the OMP, MPO and MPOMP inverses are derived as well as some properties of projectors determined by these new inverses. We establish maximal classes of matrices for which the representations of composite outer inverses are valid. Also, the integral and limit representations for OMP, MPO and MPOMP inverses are investigated. Possible applications of composite outer inverses are given too and interesting topics for further research are considered. Key words: Outer inverse, Moore-Penrose inverse, DMP inverse, CMP inverse, integral representation, limit representation

Eagle perching optimizer for the online solution of constrained optimization

The paper proposes a novel nature-inspired optimization technique called Eagle Perching Optimizer (EPO). It is an addition to the family of swarm-based meta-heuristic algorithms. It mimics eagles’ perching nature to find prey (food). The EPO is based on the exploration and exploitation of an eagle when it descends from the height such that it formulates its trajectory in a way to get to the optimal solution (prey). The algorithm takes bigger chunks of search space and looks for the optimal solution. The optimal solution in that chunk becomes the search space for the next iteration, and this process is continuous until EPO converges to the optimal global solution. We performed the theoretical analysis of EPO, which shows that it converges to the optimal solution. The simulation includes three sets of problems, i.e., uni-model, multi-model, and constrained real-world problems. We employed EPO on the benchmark problems and compared the results with state-of-the-art meta-heuristic algorithms. For the real-world problems, we used a cantilever beam, three-bar truss, and gear train problems to test the robustness of EPO and later made the comparison. The comparison shows that EPO is comparable with other known meta-heuristic algorithms