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m-weak group MP inverse

Wanlin Jiang, Jiale Gao, Xiangyu Zhang, Shengxi Zuo

Abstract

In this paper, we introduce a new matrix decomposition called the m-Core-nilpotent decomposition which is an extension of the Core-nilpotent decomposition. By this new decomposition, we propose a new generalized inverse named the m-weak group MP inverse which unifies the DMP-inverse and weak core inverse. Some characterizations, properties and representations of the m-weak group MP inverse are presented. In addition, the proposed generalized inverse is applicable to solving a restricted matrix equation.

m-weak group MP inverse

Abstract

In this paper, we introduce a new matrix decomposition called the m-Core-nilpotent decomposition which is an extension of the Core-nilpotent decomposition. By this new decomposition, we propose a new generalized inverse named the m-weak group MP inverse which unifies the DMP-inverse and weak core inverse. Some characterizations, properties and representations of the m-weak group MP inverse are presented. In addition, the proposed generalized inverse is applicable to solving a restricted matrix equation.

Paper Structure

This paper contains 7 sections, 20 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $m$-core-nilpotent decomposition and $m$-weak group MP inverse
  4. Characterizations of the $m$-WGMP inverse
  5. Representations of the $m$-WGMP inverse
  6. Applications of the $m$-WGMP inverse
  7. Conclusions

Key Result

Lemma 2.1

flt3 Let $A\in\mathbb{C}^{n\times n}_{k}$. Then the following statements are valid: $(a)$$A^{D}=A^{(2)}_{\mathcal{R}{(A^{k})},\mathcal{N}{(A^{k})}};$$(b)$$AA^{D}=A^{D}A=P_{\mathcal{R}{(A^{k})},\mathcal{N}{(A^{k})}}.$

Theorems & Definitions (42)

  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3
  • ...and 32 more