Working Paper Short Note Version 1 This version is not peer-reviewed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

## Keywords

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

## Subject

Computer Science and Mathematics, Mathematics

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