Working Paper Short Note Version 1 This version is not peer-reviewed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.

Subject Areas

range; rank; matrix product; Moore--Penrose inverse; reverse order law

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