Dual $r$-Rank Decomposition and Its Applications

TL;DR

This paper introduces the dual r-rank decomposition for dual matrices, explores its properties, and applies it to characterize dual EP matrices and generalized inverses, advancing theoretical understanding in dual matrix analysis.

Contribution

It presents the first comprehensive study of dual r-rank decomposition, including existence conditions, equivalent forms, and applications to dual Moore-Penrose inverses and dual EP matrices.

Findings

Derived existence conditions for dual r-rank decomposition.

Characterized dual EP matrices using dual r-rank decomposition.

Established relationships among dual Penrose equations.

Abstract

In this paper, we introduce the dual $r$-rank decomposition of dual matrix, get its existence condition and equivalent form of the decomposition, as well as derive some characterizations of dual Moore-Penrose generalized inverse(DMPGI). Based on DMPGI, we introduce one special dual matrix(dual EP matrix). By applying the dual $r$-rank decomposition we derive several characterizations of dual EP matrix, dual idempotent matrix, dual generalized inverses, and the relationships among dual Penrose equations.