The Dual Index and Dual Core Generalized Inverse

TL;DR

This paper introduces the dual index and dual core generalized inverse (DCGI), providing characterizations, formulas, and relationships with other inverses, and illustrating their application in linear dual equations.

Contribution

It defines the dual index and DCGI, establishes their properties, and explores their connections with existing generalized inverses, advancing the theory of dual matrix inverses.

Findings

DCGI exists if and only if the dual index of the matrix is one.

The paper provides a compact formula for DCGI and characterizations of its existence.

Examples demonstrate the application of DCGI in solving linear dual equations.

Abstract

In this paper, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse and matrix decomposition, we give several characterizations of the dual index when it is equal to one. And we get that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth nothing that the dual index of $\hat{M}$ is equal to one if and only if its DCGI exists. When the dual index of $\hat{M}$ is equal to one, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI), and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse (MPDGI) and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. At last, two examples are given to illustrate the application of DCGI in linear dual equations.