Abstract
The generalized inverse has many important applications in the aspects of theoretic research of matrices and statistics. One of the core problems in generalized inverse is to find the necessary and sufficient conditions of the forward order laws for generalized inverse of matrix product. In this paper, by using the expressions for maximal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws \(A_1\{1,3\}A_2\{1,3\}A_3\{1,3\}\subseteq (A_1A_2A_3)\{1,3\}\) and \(A_1\{1,4\}A_2\{1,4\}A_3\{1,4\}\subseteq (A_1A_2A_3)\{1,4\}\).
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Acknowledgements
The authors would like to thank Professor Orizon Pereira Ferreira and the anonymous referees for their very detailed comments and constructive suggestions, which greatly improved the presentation of this paper.
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Communicated by Orizon Pereira Ferreira.
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This work was supported by the National Natural Science Foundation of China (nos: 11771159, 11571004) and the Natural Science Foundation of GuangDong (no: 2014A030313625) and the Training plan for the Outstanding Young Teachers in Higher Education of Guangdong (no: SYq2014002).
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Liu, Z., Xiong, Z. & Qin, Y. A note on the forward order law for least square g-inverse of three matrix products. Comp. Appl. Math. 38, 48 (2019). https://doi.org/10.1007/s40314-019-0803-y
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DOI: https://doi.org/10.1007/s40314-019-0803-y