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The W-weighted m-weak group MP inverse and its applications

Jiale Gao, Kezheng Zuo, Qing-Wen Wang

Abstract

We extend the concept of the m-weak group MP inverse of a square matrix to a rectangular matrix, called the W-weighted m-weak group MP inverse, which also unifies the W-weighted weak core inverse and W-weighted DMP inverse. Some properties, characterizations and representations of this new generalized inverse are shown. Additionally, applications of the W-weighted weak group MP inverse are given in solving a constrained optimization problem and a class of consistent matrix equations.

The W-weighted m-weak group MP inverse and its applications

Abstract

We extend the concept of the m-weak group MP inverse of a square matrix to a rectangular matrix, called the W-weighted m-weak group MP inverse, which also unifies the W-weighted weak core inverse and W-weighted DMP inverse. Some properties, characterizations and representations of this new generalized inverse are shown. Additionally, applications of the W-weighted weak group MP inverse are given in solving a constrained optimization problem and a class of consistent matrix equations.

Paper Structure

This paper contains 5 sections, 22 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $W$-weighted $m$-weak group MP inverse
  4. Applications of the $W$-weighted $m$-weak group MP inverse
  5. Conclusions

Key Result

Lemma 2.1

FerreyraWeCoerEPinref Let $A\in\mathbb{C}^{q\times n}$, $W(\neq 0)\in\mathbb{C}^{n\times q}$ and $k=\max\{{\rm Ind}(AW),{\rm Ind}(WA)\}$. Then, where $U\in\mathbb{C}^{q\times q}$ and $V\in\mathbb{C}^{n\times n}$ are unitary matrices, $A_1\in\mathbb{C}^{t\times t }$ and $W_1\in\mathbb{C}^{t\times t }$ are nonsingular matrices, $A_2\in\mathbb{C}^{t\times (n-t)}$, $W_2\in\mathbb{C}^{t\times (q-t)}$,

Theorems & Definitions (38)

  • Lemma 2.1: Weighted core-EP decomposition
  • Lemma 2.2: Weighted Hartwig-Spindelböck decomposition
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4
  • ...and 28 more