The reverse order laws for $ \{1, 2, 3M\} $- and $ \{1, 2, 4N\} $- inverse of three matrix products

Baifeng Qiu; Yingying Qin; Zhiping Xiong; Baifeng Qiu; Yingying Qin; Zhiping Xiong

doi:10.3934/math.2025033

AIMS Mathematics

2025, Volume 10, Issue 1: 721-735. doi: 10.3934/math.2025033

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The reverse order laws for $ \{1, 2, 3M\} $- and $ \{1, 2, 4N\} $- inverse of three matrix products

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, Guangdong, China

Received: 24 October 2024 Revised: 16 December 2024 Accepted: 03 January 2025 Published: 13 January 2025
MSC : 15A09, 15A24, 47A05

The reverse order laws for weighted generalized inverses often appear in linear algebra problems of several applied fields, having attracted considerable attention. In this paper, by using the maximal and minimal ranks of the generalized Schur complement, we obtained some necessary and sufficient conditions for the reverse order laws
$ A_3\{1,2,3M_3\}A_2\{1,2,3M_2\}A_1\{1,2,3M_1\}\subseteq (A_1A_2A_3)\{1,2,3M_1\} $
and
$ A_3\{1,2,4N_{4}\}A_{2}\{1,2,4N_{3}\}A_1\{1,2,4N_2\}\subseteq (A_1A_2 A_3)\{1,2,4N_{4}\}. $
- reverse order law,
- weighted generalized inverses,
- generalized Schur complement,
- maximal and minimal ranks,
- matrix product
Citation: Baifeng Qiu, Yingying Qin, Zhiping Xiong. The reverse order laws for $ \{1, 2, 3M\} $- and $ \{1, 2, 4N\} $- inverse of three matrix products[J]. AIMS Mathematics, 2025, 10(1): 721-735. doi: 10.3934/math.2025033

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Abstract

The reverse order laws for weighted generalized inverses often appear in linear algebra problems of several applied fields, having attracted considerable attention. In this paper, by using the maximal and minimal ranks of the generalized Schur complement, we obtained some necessary and sufficient conditions for the reverse order laws

$ A_3\{1,2,3M_3\}A_2\{1,2,3M_2\}A_1\{1,2,3M_1\}\subseteq (A_1A_2A_3)\{1,2,3M_1\} $

and

$ A_3\{1,2,4N_{4}\}A_{2}\{1,2,4N_{3}\}A_1\{1,2,4N_2\}\subseteq (A_1A_2 A_3)\{1,2,4N_{4}\}. $

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