曹志强 杜乃林

摘 要： 研究了两个Lp.s.d.矩阵广义BottDuffin逆的逆序律问题，Lp.s.d.是半正定概念的一种推广，正定或半正定矩阵恰是R np.s.d.矩阵.通过建立广义BottDuffin逆与A（2）S，S⊥逆之间的关系，利用A（2）S，S⊥的性质，得到了广义BottDuffin逆序律成立的充要条件，并通过两个实例验证了所得结果.

关键词： 逆序律; Lp.s.d.矩阵; 广义BottDuffin逆

中图分类号： O 151.21  文献标志码： A  文章编号： 10005137（2019）04034709

Abstract： In this paper we investigate the reverse order law for the generalized BottDuffin inverses of two Lp.s.d.matrices.The concept of Lp.s.d.matrices is a generalization of semipositive definiteness，in the sense that Rnp.s.d.matrices are exactly semipositive definite matrices.We establish a relationship between generalized BottDuffin inverses and A（2）S，S⊥ inverses，and by using their properties，give necessary and sufficient conditions for the reverse order law of the generalized BottDuffin inverses.In the end we give two explicit examples to test our results.

Key words： reverse order law; Lp.s.d.matrix; generalized BottDuffin inverse

参考文献：

[1] PENROSE R.A Generalized Inverse for Matrices [M].Cambridge：Cambridge University Press，1955：406-413.

[2] GREVILLE T N E.Note on the generalized inverse of a matrix product [J].Siam Review，1966，8（4）：518-521.

[3] HARTWIG R E.The reverse order law revisited [J].Linear Algebra and Its Applications，1986，76：241-246.

[4] TIAN Y.Reverse order laws for the generalized inverses of multiple matrix products [J].Linear Algebra and Its Applications，1994，211：85-100.

[5] SHINOZAKI N，SIBUYA M.The reverse order law （AB）- = B- A- [J].Linear Algebra and Its Applications，1974，9：29-40.

[6] WEI M.Reverse order laws for generalized inverses of multiple matrix products [J].Linear Algebra and Its Applications，1999，293（1/2/3）：273-288.

[7] ZHENG B，XIONG Z P.The reverse order laws for 1，2，3and 1，2，4inverses of multiple matrix products [J].Linear and Multilinear Algebra，2010，58（6）：765-782.

[8] TIAN H J.On the reverse order law （AB）D= BDAD [J].Journal of Mathematical Research Exposition，1999，19：355-358.

[9] CAO C，ZHANG X，TANG X.Reverse order law of group inverses of products of two matrices [J].Applied Mathematics and Computation，2004，158（2）：489-495.

[10] WANG G，WEI Y，QIAO S.Generalized Inverses：Theory and Computations [M].Beijing：Science Press，2004：113-211.

[11] BOTT R，DUFFIN R J.On the algebra of networks [J].Transactions of the American Mathematical Society，1953，74（1）：99-109.

[12] GREVILLE T N E，BENISRAEL A.Generalized Inverses：Theory and Applications [M].New York：Wiley，1974：112-115.

[13] CHEN Y L.The generalized BottDuffin inverse and its applications [J].Linear Algebra and Its Applications，1990，134：71-91.

[14] YU Y M，WANG G R.The existence of the generalized inverse A（2）T，S of a matrix A over skew fields [J].Journal of Shanghai Normal University （Natural Sciences），2007，36（2）：1-5.

[15] YU Y，WANG G.The generalized inverse A（2）T，S of a matrix over an associative ring [J].Journal of the Australian Mathematical Society，2007，83（3）：423-438.

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