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On Rayleigh Quotient Iteration for Dual Quaternion Hermitian Eigenvalue Problem

Shan-Qi Duan, Qing-Wen Wang, Xue-Feng Duan

Abstract

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing minimal residual property of the Rayleigh Quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.

On Rayleigh Quotient Iteration for Dual Quaternion Hermitian Eigenvalue Problem

Abstract

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing minimal residual property of the Rayleigh Quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.
Paper Structure (11 sections, 16 theorems, 100 equations, 2 figures, 2 tables, 2 algorithms)

This paper contains 11 sections, 16 theorems, 100 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Dual Quaternions and Dual Quaternion Matrices
  3. Dual numbers, quaternions and dual quaternions
  4. Dual quaternion matrices
  5. Dual Representation of a Dual Quaternion Matrix
  6. Rayleigh Quotient Iteration for Computing the Eigenvalue of a Dual Quaternion Hermitian Matrix
  7. On the solution of the linear system $(\hat{A}-\hat{\theta} I)\hat{\mathbf{w}}=\hat{\mathbf{u}}$
  8. Convergence Analysis
  9. Computing All Appreciable Eigenvalues of a Dual Quaternion Hermitian Matrix
  10. Numerical Experiments
  11. Conclusion

Key Result

Lemma 2.1

Given an appreciable dual quaternion vector $\hat{\mathbf{x}}\in \mathbb{DQ}^{n}$, the norm $\|\cdot\|_{2}$ is induced by the inner product, i.e. $\|\hat{\mathbf{x}}\|_{2}=\sqrt{\langle\hat{\mathbf{x}},\hat{\mathbf{x}}\rangle}$.

Figures (2)

  • Figure 1: The convergence curves of RQI for computing all appreciable eigenvalues of the dual quaternion Hermitian matrix $\hat{A}$.
  • Figure 2: The convergence curve of RQI and PM for computing an eigenvalue with the random Laplacian matrices.

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • ...and 20 more