Research On Reverse Order Law For Weighted Generalized Inverse

In recent decades, with a wide application of the generalized inverse in math-ematical theory and practice, the problem of the reverse order law for generalized inverse becomes a valuable investigated direction in the theory of the generalized in-verse. In other words, it is worthwhile doing some researches on the conditions in which the generalized inverse of matrices product has the similar property as the reg-ular inverse. This paper mainly researches the reverse order law for the weighted Moore-Penrose inverse of the product of two and three matrices.Firstly, based on Yonglin Chen's work in [2], we derive the detailed proof of the necessary and sufficient conditions of the reverse order law for the weighted Moore-Penrose inverse of the product of two and three matrices respectively. Secondly, ac-cording to Wenyu Sun and Yimin Wei's methods in [40], we provide a new proof of the necessary and sufficient conditions of the reverse order law for the weighted Moore-Penrose inverse of the product of two and three matrices. In addition, some new necessary and sufficient conditions are also given. Thirdly, a simple rank condi-tion method is used to establish a class of identities for the weighted Moore-Penrose inverse of the product of two and three matrices. For the weighted Moore-Penrose inverse of the product of two matrices, this paper gives 5 pairs of the matrices E and F which respectively satisfy the rank conditions (4.1.1) and (4.1.2), and therefore, we can get 35 identities of (AB)MP in all, and we also give proofs of 15 identities thereof. Similarly, for the weighted Moore-Penrose inverse of the product of three matrices, this paper gives 6 pairs of the matrices E and F which respectively satisfy the rank conditions (4.2.1) and (4.2.2), and therefore, we can get 48 identities of (ABC)(?)MQ in all, and we also give proofs of 18 identities thereof. Finally, this paper introduces some corresponding identities when M=I_m, N=I_n, P=I_p, Q=I_q. The results show that many existing conclusions are a special case of this paper. Hence, by apply-ing this method, we can not only obtain many known identities in a more simple way, but also derive a lot of new identities simply and directly. In the end of this paper, we give some follow-up research questions.