Short Note
Version 1
This version is not peer-reviewed
Establishment and Characterization of Equalities for the Ranges of Matrices
Version 1
: Received: 23 March 2021 / Approved: 23 March 2021 / Online: 23 March 2021 (09:56:59 CET)
How to cite: Tian, Y. Establishment and Characterization of Equalities for the Ranges of Matrices. Preprints 2021, 2021030561 Tian, Y. Establishment and Characterization of Equalities for the Ranges of Matrices. Preprints 2021, 2021030561
Abstract
This note addresses a fundamental problem in matrix theory on establishing and characterizing range equalities for matrix expressions that involve generalized inverses. We first establish a group of necessary and sufficient conditions for the matrix range equality ${\rm range}(D_1 - C_1A_1^{\dag}B_1) = {\rm range}(D_2 - C_2A_2^{\dag}B_2)$ to hold, where $(\cdot)^{\dag}$ denotes the Moore--Penrose inverse of matrix. We then give several groups of range equalities with extrusion properties for multiple matrix products associated with two matrices and their conjugate transposes and Moore--Penrose inverses.
Keywords
range; rank; matrix product; Moore--Penrose inverse; reverse order law
Subject
Computer Science and Mathematics, Mathematics
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.
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