In this article, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse, and matrix decomposition, we give several characterizations of ...the dual index when it is equal to 1. We realize that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth noting that the dual index of
is equal to 1 if and only if its DCGI exists. When the dual index of
is equal to 1, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI) and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse, and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. Finally, two examples are given to illustrate the application of DCGI in linear dual equations.
Recently, the dual Moore-Penrose generalized inverse has been applied to study the linear dual equation when the dual Moore-Penrose generalized inverse of the coefficient matrix of the linear dual ...equation exists. Nevertheless, the dual Moore-Penrose generalized inverse exists only in partially dual matrices. In this paper, to study more general linear dual equation, we introduce the weak dual generalized inverse described by four dual equations, and is a dual Moore-Penrose generalized inverses for it when the latter exists. Any dual matrix has the weak dual generalized inverse and is unique. We obtain some basic properties and characterizations of the weak dual generalized inverse. Also, we investigate relationships among the weak dual generalized inverse, the Moore-Penrose dual generalized inverses and the dual Moore-Penrose generalized inverses, give the equivalent characterization and use some numerical examples to show that the three are different dual generalized inverse. Afterwards, by applying the weak dual generalized inverse we solve two special linear dual equations, one of which is consistent and the other is inconsistent. Neither of the coefficient matrices of the above two linear dual equations has dual Moore-Penrose generalized inverses.
•If a Moore-Penrose dual Inverse of a matrix exists, it must be unique.•Necessary and sufficient conditions for a {1}-inverse of a matrix to exist.•Necessary and sufficient conditions for a system of ...linear dual equations to have a solution, and explicit solutions for such a system of equations.•Dual analog to finding Least-Squares solutions to a system of real equations•Dual analog of finding Least squares Minimum-length solution to a system of real equations.•Applications to kinematics.
This paper deals with properties of dual generalized inverses and uses them in solving linear dual equations. The first part of this paper addresses several properties of dual generalized inverses of dual matrices and provides new theoretical and computational insights. In the second part necessary and sufficient conditions for a dual matrix to have a {1}-dual generalized inverse are obtained. A dual matrix with a rank deficient primal part generically (almost always) is shown to have no {1}-dual generalized inverse and hence, neither a {1,3}-dual generalized inverse or a Moore-Penrose dual generalized inverse. The necessary and sufficient condition to have a solution to a system of linear dual equations is obtained. An explicit expression for the solution, when one exists, is obtained and it is shown not to be unique, in general. The dual analog of the real least-squares problem is obtained through the {1,3}-dual generalized inverse; as with real matrices, it is shown that this least-squares dual solution is not unique, in general. The dual analog of the real minimum-length least-squares solution is obtained through the development of the Moore-Penrose dual generalized inverse. Numerical examples are provided throughout, mainly from kinematics, to illustrate the results obtained.
In this article, we study the m-WG∘ inverse which presents a generalization of the m-WG inverse in the Minkowski space. We first show the existence and the uniqueness of the generalized inverse. ...Then, we discuss several properties and characterizations of the m-WG∘ inverse by using the core-EP decomposition. Applying the generalized inverse, we obtain the solutions of some matrix equations in Minkowski space.
In this paper, we revise the core EP inverse of a square matrix introduced by Prasad and Mohana in 12, Core EP inverse, Linear and Multilinear Algebra 62(3) (2014), 792-802. Firstly, we give a new ...representation and a new characterization of the core EP inverse. Then, we study some properties of the core EP inverse by using a representation by block matrices. Secondly, we extend the notion of core EP inverse to rectangular matrices by means of a weighted core EP decomposition. Finally, we study some properties of weighted core EP inverses.
We calculate the ratio Rℓℓ of same sign (SS) to opposite sign (OS) dileptons in type I and generalized inverse seesaw models and show that it can be anywhere between 0 and 1 depending on the detailed ...texture of the right-handed neutrino mass matrix. Measurement of Rℓℓ in hadron colliders can therefore provide a way to probe the nature of seesaw mechanism and also to distinguish between the two types of seesaw mechanisms. We work within the framework of left-right symmetric model as an example. We emphasize that coherence of the final states in the WR decay is crucial for this discussion and it requires the right-handed neutrinos to be highly degenerate. We isolate the range of parameters in the model where this effect is observable at the LHC and future colliders.
•Necessary and suffiucient conditions for a dual matrix to have the DMPGI.•Necessary and sufficient conditions for the MPDGI to be the DMPGI.•A compact formula for the computation of the DMPGI.
In ...this paper, we give necessary and sufficient conditions for the dual matrix M^ to have the dual Moore-Penrose generalized inverse (for short DMPGI) and give necessary and sufficient conditions for the Moore-Penrose dual generalized inverse (for short MPDGI) of M^ to be a DMPGI of M^. At last, a compact formula for the computation of the DMPGI is given.
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