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Properties of core-EP matrices and binary relationships

Ehsan Kheirandish, Abbas Salemi, Néstor Thome

TL;DR

The paper investigates core-EP matrices through the lens of the MPDMP matrix $A^{\dagger ,d,\dagger}$ and related inverses. It proves that for core-EP matrices the DMP, MPD, and Drazin inverses coincide, and it develops multiple equivalent characterizations of core-EP via $A^{\dagger ,d,\dagger}$, the core part $A_c$, and projection relations $Q_A$ and $P_A$. It also derives Greville-type formulas for $A^{d,\dagger}$ and $A^{\dagger ,d}$, and analyzes CMP, DMP, MPD inverses and their interactions within the core-EP framework. Finally, it introduces DMP/MPD binary relations and shows how the core part bounds these orders and when these relations align (notably for $k$-EP matrices), providing a unified view of generalized inverses and matrix orders in core-EP theory.

Abstract

In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with $A$ and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.

Properties of core-EP matrices and binary relationships

TL;DR

The paper investigates core-EP matrices through the lens of the MPDMP matrix and related inverses. It proves that for core-EP matrices the DMP, MPD, and Drazin inverses coincide, and it develops multiple equivalent characterizations of core-EP via , the core part , and projection relations and . It also derives Greville-type formulas for and , and analyzes CMP, DMP, MPD inverses and their interactions within the core-EP framework. Finally, it introduces DMP/MPD binary relations and shows how the core part bounds these orders and when these relations align (notably for -EP matrices), providing a unified view of generalized inverses and matrix orders in core-EP theory.

Abstract

In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.
Paper Structure (4 sections, 32 theorems, 79 equations)

This paper contains 4 sections, 32 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Properties of core-EP matrices
  3. Some properties of CMP, DMP and MPD inverses
  4. DMP and MPD binary relationships

Key Result

theorem 1

Let $A \in M_{n}(\mathbb{C})$. Then $X= A^{d}A^{\dagger}$ is the unique solution of the following equations: Analogously, the unique matrix that satisfies is given by $X=A^{\dagger}A^{d}$.

Theorems & Definitions (58)

  • theorem 1
  • proof
  • theorem 2
  • proof
  • definition 1
  • remark 1
  • theorem 3
  • proof
  • corollary 1
  • proposition 1
  • ...and 48 more