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LU Decomposition and Generalized Autoone-Takagi Decomposition of Dual Matrices and their Applications

Renjie Xu, Yimin Wei, Hong Yan

TL;DR

The paper addresses the extension of classical matrix decompositions to dual-number and dual-quaternion settings, focusing on Autonne-Takagi decompositions for dual complex symmetric and dual quaternion $\eta$-Hermitian matrices, as well as LU and Cholesky factorizations for dual matrices. It develops a Sylvester-equation framework to derive the dual LU decomposition (DLU) and establishes its equivalence to dual rank-$k$ decompositions and the dual Moore-Penrose generalized inverse (DMPGI). It further generalizes Takagi-type factorizations to the dual-quaternion domain (nontrivial through $\eta$-Hermitian structure) and provides algorithms/pseudocode for the Autonne-Takagi-type decompositions and ATDSVD, along with dual Cholesky results for dual real SPD matrices. The experimental section demonstrates the method on dual Hankel and SPD-like matrices, illustrating error behavior and numerical stability, and underscores the practical relevance to numerical linear algebra in dual-number contexts.

Abstract

This paper uses matrix transformations to provide the Autoone-Takagi decomposition of dual complex symmetric matrices and extends it to dual quaternion $η$-Hermitian matrices. The LU decomposition of dual matrices is given using the general solution of the Sylvester equation, and its equivalence to the existence of rank-k decomposition and dual Moore-Penrose generalized inverse (DMPGI) is proved. Similar methods are then used to provide the Cholesky decomposition of dual real symmetric positive definite matrices. Both of our decompositions are driven by applications in numerical linear algebra.

LU Decomposition and Generalized Autoone-Takagi Decomposition of Dual Matrices and their Applications

TL;DR

The paper addresses the extension of classical matrix decompositions to dual-number and dual-quaternion settings, focusing on Autonne-Takagi decompositions for dual complex symmetric and dual quaternion -Hermitian matrices, as well as LU and Cholesky factorizations for dual matrices. It develops a Sylvester-equation framework to derive the dual LU decomposition (DLU) and establishes its equivalence to dual rank- decompositions and the dual Moore-Penrose generalized inverse (DMPGI). It further generalizes Takagi-type factorizations to the dual-quaternion domain (nontrivial through -Hermitian structure) and provides algorithms/pseudocode for the Autonne-Takagi-type decompositions and ATDSVD, along with dual Cholesky results for dual real SPD matrices. The experimental section demonstrates the method on dual Hankel and SPD-like matrices, illustrating error behavior and numerical stability, and underscores the practical relevance to numerical linear algebra in dual-number contexts.

Abstract

This paper uses matrix transformations to provide the Autoone-Takagi decomposition of dual complex symmetric matrices and extends it to dual quaternion -Hermitian matrices. The LU decomposition of dual matrices is given using the general solution of the Sylvester equation, and its equivalence to the existence of rank-k decomposition and dual Moore-Penrose generalized inverse (DMPGI) is proved. Similar methods are then used to provide the Cholesky decomposition of dual real symmetric positive definite matrices. Both of our decompositions are driven by applications in numerical linear algebra.
Paper Structure (14 sections, 9 theorems, 62 equations, 5 algorithms)

This paper contains 14 sections, 9 theorems, 62 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual Numbers
  4. Quaternions
  5. Autonne-Takagi Decomposition
  6. Autonne-Takagi Decomposition of Dual Complex Symmetric Matrices (DTakagi)
  7. Dual Quaternion .-Hermitian Matrix
  8. LU Decomposition of Dual Matrices (DLU)
  9. DMPGI and Rank of Dual Matrices
  10. Dual Cholesky Decomposition
  11. Experiments
  12. Conclusion
  13. Data availability
  14. Acknowledgements

Key Result

Theorem 3.1

(DTakagi) For a dual complex symmetric matrix $A = A_{s} + \epsilon A_{i} \in \mathbb{DC}^{n\times n}$, we can perform the decomposition as follows, where $\Sigma = \Sigma_{s}+\epsilon \Sigma_{i}\in \mathbb{DR}^{n\times n}$ is a dual diagonal matrix, and $V = V_{s}+\epsilon V_{i}\in \mathbb{DC}^{n\times n}$ is unitary.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Definition 3.1
  • Theorem 3.2: horn2012generalization, Theorem 3
  • Theorem 3.3
  • Remark 3.1
  • Lemma 4.1
  • Theorem 4.1
  • Remark 4.1
  • ...and 8 more