Aequationes mathematicae

, Volume 73, Issue 1–2, pp 56–70 | Cite as

On reverse-order laws for least-squares g-inverses and minimum norm g-inverses of a matrix product

  • Yoshio Takane
  • Yongge Tian
  • Haruo Yanai


Let A(1,3) and A(1,4) denote a least-squares g-inverse and a minimum norm g-inverse of a matrix A, respectively. In this paper, we establish necessary and sufficient conditions for \({\{B^{(1,3)}A^{(1,3)}\} \subseteq \{(AB)^{(1,3)}\}}\) and \({\{B^{(1,4)}A^{(1,4)}\} \subseteq \{(AB)^{(1,4)}\}}\) to hold. We also show that the well-known reverse-order law \({(AB)^{\dagger}= B^{\dagger} A^{\dagger}}\) is equivalent to \({\{B^{(1,3)}A^{(1,3)}\} \subseteq \{(AB)^{(1,3)}\}}\) and \({\{B^{(1,4)}A^{(1,4)}\} \subseteq \{(AB)^{(1,4)}\}}\) .

Mathematics Subject Classification (2000).

15A03 15A09 


Least-squares g-inverse matrix product matrix rank method minimum norm g-inverse Moore-Penrose inverse reverse-order law 


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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Department of PsychologyMcGill UniversityMontréalCanada
  2. 2.School of EconomicsShanghai University of Finance and EconomicsShanghaiChina
  3. 3.St. Luke’s College of NursingTokyoJapan

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