Twists of two or multiple idempotent matrices

TL;DR

This paper explores block matrix methods to derive rank formulas for matrix expressions involving idempotent matrices, with applications in generalized inverse characterizations.

Contribution

It introduces new rank expansion formulas for idempotent matrix expressions and applies them to characterize generalized inverses of partitioned matrices.

Findings

Derived new rank equalities for idempotent matrix expressions

Provided applications in characterizing generalized inverses

Enhanced understanding of matrix rank calculations

Abstract

In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank equalities for matrix expressions composed by idempotent matrices, and present their applications in the characterization of some matrix equalities for generalized inverses of partitioned matrices.