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In this paper we extend the definition of BT inverse of square complex matrices to rectangular matrices. This new generalized inverse (called the W -weighted BT inverse) always exists and is unique. We investigate several characterizations of the W -weighted BT inverse. We give two representations of the W -weighted BT inverse; the first one uses the weighted core EP decomposition and the second one is given in terms of a singular value decomposition. In addition, we derive some properties of the W -weighted BT inverse and its relationship with others well-known inverses. [ABSTRACT FROM AUTHOR]

In this paper, we present several representations of the W-weighted WG inverse. These representations are expressed in terms of matrix powers as well as in terms of matrix products involving only the Moore–Penrose inverse. In addition, a new characterization of the W-weighted WG inverse is presented by using a rank equation. [ABSTRACT FROM AUTHOR]

This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations and representations of GDMP-inverses are obtained by means of the core-nilpotent and the Hartwig-Spindelböck decompositions. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary and sufficient conditions for the k-commutative equality A k X = X A k , where X is an outer generalized inverse of the square matrix A. Also, we give new representations for core EP, DMP, and CMP inverses of square matrices as outer inverses with prescribed null space and range. In addition, we characterize the core EP inverse as the solution of a new system of matrix equations. [ABSTRACT FROM AUTHOR]

In this paper, {1}-inverses of a nilpotent matrix as well as matrices above a given nilpotent matrix under the minus partial order are characterized. [ABSTRACT FROM AUTHOR]

This paper deals with singular systems of index. Our main goal is to find a state-feedback such that the closed-loop system satisfies the regularity condition and it is nonnegative and stable. In order to do that, the core-nilpotent decomposition of a square matrix is applied to the singular matrix of the system. Moreover, if the Drazin projector of this matrix is nonnegative, then the previous decomposition allows us to write the core-part of the matrix in a specific block form. In addition, an algorithm to study this kind of systems via a state-feedback is designed. [ABSTRACT FROM AUTHOR]

This paper derives some further results on recent generalized inverses studied in the literature, namely core EP, DMP, and CMP inverses. Our main aim is to develop maximal classes of matrices for which their representations remain valid. [ABSTRACT FROM AUTHOR]

This paper deals with weighted G-Drazin inverses, which is a new class of matrices introduced to extend (to the rectangular case) G-Drazin inverses recently considered by Wang and Liu for square matrices. First, we define and characterize weighted G-Drazin inverses. Next, we consider a new pre-order defined on complex rectangular matrices based on weighted G-Drazin inverses. Finally, we characterize this pre-order and relate it to the minus partial order and to the weighted Drazin pre-order. [ABSTRACT FROM AUTHOR]

In this paper, we investigate new binary relations defined on the set of rectangular complex matrices based on the weighted Drazin inverse and give some characterizations of them. These relations become pre-orders and improve the results found by the authors in Hernández et al. (2013) as well as extend those known for square matrices. On the other hand, some new weighted partial orders are also defined and characterized. The advantages of these new relations compared to the ones considered in the mentioned paper are also pointed out. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal { k + 1 } -potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained. [ABSTRACT FROM AUTHOR]