Version 1
: Received: 19 February 2023 / Approved: 20 February 2023 / Online: 20 February 2023 (07:48:05 CET)
How to cite:
Tian, Y. Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses. Preprints2023, 2023020329. https://doi.org/10.20944/preprints202302.0329.v1
Tian, Y. Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses. Preprints 2023, 2023020329. https://doi.org/10.20944/preprints202302.0329.v1
Tian, Y. Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses. Preprints2023, 2023020329. https://doi.org/10.20944/preprints202302.0329.v1
APA Style
Tian, Y. (2023). Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses. Preprints. https://doi.org/10.20944/preprints202302.0329.v1
Chicago/Turabian Style
Tian, Y. 2023 "Constructions of Mixed Reverse-Order Laws for the Products of Matrices Involving Moore–Penrose Inverses and Group Inverses" Preprints. https://doi.org/10.20944/preprints202302.0329.v1
Abstract
It is a classic topic in algebras to construct and verify equalities that are composed of various algebraic operations of elements and their inverses or generalized inverses. In this note, the author constructs a matrix equality $(AB)^{\dag} = B^{\ast}(A^{\ast}ABB^{\ast})^{\#}A^{\ast}$ (called a mixed reverse-order law), where $A$ and $B$ are two matrices of appropriate sizes, $(\cdot)^{\ast}$, $(\cdot)^{\dag}$, and $(\cdot)^{\#}$ denote the conjugate transpose, the Moore--Penrose inverse, and the group inverse of a matrix, respectively, and shows that the mixed reverse-order law always holds. A list of variation forms of this matrix equality are also given, and necessary and sufficient conditions are obtained for them to hold. Especially, we show a remarkable fact that the two reverse-order laws $(AB)^{\dag} = B^{\dag}A^{\dag}$ and $(A^{\ast}ABB^{\ast})^{\#} = (BB^{\ast})^{\#}(A^{\ast}A)^{\#}$ are equivalent.
Keywords
block matrix; group inverse; Moore--Penrose inverse; range; rank; reverse-order law
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
