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Further results for the dual Hartwig-Spindelb{ö}ck decomposition and its applications

Tan Mei, Kezheng Zuo, Hui Yan

TL;DR

This work advances the theory of dual matrices by introducing two new forms of the dual Hartwig–Spindelböck decomposition (D-H–S) and deriving explicit representations for several dual generalized inverses. Building on dual-SVD, it analyzes the relationships among inverses, develops composite inverse forms, and demonstrates applicability to dual partial orders. The results yield a systematic framework for dual matrix analysis with practical formulas for the dual Moore–Penrose inverse and related inverses under various structural conditions. The approach offers a compact, algebraic toolset for robotics, motion analysis, and other areas where dual numbers encode size, direction, and position.

Abstract

In this paper, we introduce two new forms of the dual Hartwig-Spindelb{ö}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further explore and characterize the relationships and properties of these inverses, investigate the dual composite generalized inverses, and verify the applicability of dual partial orders. The proposed decomposition provides a systematic and convenient framework for the study of dual matrices.

Further results for the dual Hartwig-Spindelb{ö}ck decomposition and its applications

TL;DR

This work advances the theory of dual matrices by introducing two new forms of the dual Hartwig–Spindelböck decomposition (D-H–S) and deriving explicit representations for several dual generalized inverses. Building on dual-SVD, it analyzes the relationships among inverses, develops composite inverse forms, and demonstrates applicability to dual partial orders. The results yield a systematic framework for dual matrix analysis with practical formulas for the dual Moore–Penrose inverse and related inverses under various structural conditions. The approach offers a compact, algebraic toolset for robotics, motion analysis, and other areas where dual numbers encode size, direction, and position.

Abstract

In this paper, we introduce two new forms of the dual Hartwig-Spindelb{ö}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further explore and characterize the relationships and properties of these inverses, investigate the dual composite generalized inverses, and verify the applicability of dual partial orders. The proposed decomposition provides a systematic and convenient framework for the study of dual matrices.
Paper Structure (5 sections, 22 theorems, 104 equations)

This paper contains 5 sections, 22 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preparation
  3. D-H-S decomposition
  4. Applications of the D-H-S Decomposition
  5. Conclusion

Key Result

Lemma 2.1

(Qi and Luo Qi1) Let $\hat{A} \in \mathbb{DC}^{m \times n}$. Then there exist $\hat{U} \in \mathbb{DC}_m^{U}$ and $\hat{V} \in \mathbb{DC}_n^{U}$ such that where $\mu_{1} \geq \mu_{2} \geq \cdots \geq \mu_{r}$ are positive appreciable dual numbers, and $\mu_{r+1} \geq \mu_{r+2} \geq \cdots \geq \mu_{t}$ are positive infinitesimal dual numbers. Counting possible multiplicities of the diagonal ent

Theorems & Definitions (49)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Theorem 3.1
  • proof
  • ...and 39 more