The Forward Order Laws for \(\{1,2,3\}\)- and \(\{1,2,4\}\)-Inverses of Multiple Matrix Products

  • Zhiping XiongEmail author
  • Zhongshan Liu


The forward order law for generalized inverse often appears in linear algebra problems of some applied fields, which have attracted considerable attention and some interesting results have been obtained. In this paper, using the extremal ranks of the generalized Schur complement, we obtain the necessary and sufficient conditions for the forward order laws
$$\begin{aligned} A_1\{1,2,3\}A_2\{1,2,3\}\cdots A_n\{1,2,3\}\subseteq (A_1A_2\cdots A_n)\{1,2,3\} \end{aligned}$$
$$\begin{aligned} A_1\{1,2,4\}A_2\{1,2,4\}\cdots A_n\{1,2,4\}\subseteq (A_1A_2\cdots A_n)\{1,2,4\}. \end{aligned}$$


Forward order law Generalized inverse Maximal and minimal ranks Matrix product Generalized Schur complement 

Mathematics Subject Classification

47A05 15A09 15A24 



The authors would like to thank Professor Daniel Aron Alpay and the anonymous referees for their very detailed comments and constructive suggestions, which greatly improved the presentation of this paper.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computational ScienceWuyi UniversityJiangmenPeople’s Republic of China

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