Weak dual generalized inverse of a dual matrix and its applications.

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.