Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Characterization of a Nonlinear Equality Composed of Multiple Products of Matrices and their Generalized Inverses

Version 1 : Received: 19 June 2022 / Approved: 20 June 2022 / Online: 20 June 2022 (10:53:30 CEST)

How to cite: Tian, Y. Characterization of a Nonlinear Equality Composed of Multiple Products of Matrices and their Generalized Inverses. Preprints 2022, 2022060271. https://doi.org/10.20944/preprints202206.0271.v1 Tian, Y. Characterization of a Nonlinear Equality Composed of Multiple Products of Matrices and their Generalized Inverses. Preprints 2022, 2022060271. https://doi.org/10.20944/preprints202206.0271.v1

## Abstract

One of matrix equalities composed of multiple products of matrices and their generalized inverses is given by $A_1B_1^{-}A_2B_2^{-} \cdots A_kB_k^{-}A_{k+1}= A$ where $A_1$, $B_1$, $A_2$, $B_2$, $\ldots$, $A_k$, $B_k$, $A_{k+1}$, and $A$ are given matrices of appropriate sizes, and $B_1^{-}$, $B_2^{-}$, $\cdots$, $B_k^{-}$ are generalized inverses of matrices. The cases for $k = 1, 2$ and their special forms were properly approached in the theory of generalized inverses of matrices. In this note, the author presents an algebraic procedure to derive explicit necessary and sufficient conditions for the equality with $k = 3$ to always hold using certain rank equalities for the block matrices constructed by the given matrices, and then mention a key step of extending the previous work to a general situation.

## Keywords

generalized inverse; matrix product; matrix equality; set inclusion; rank equality; block matrix

## Subject

Computer Science and Mathematics, Algebra and Number Theory