﻿ 四元数矩阵特征值的Jacobi迭代 The Jacobi Iteration of Eigenvalue of Real Self-Adjoint Quaternion Matrices

Pure Mathematics
Vol.08 No.03(2018), Article ID:25073,5 pages
10.12677/PM.2018.83036

The Jacobi Iteration of Eigenvalue of Real Self-Adjoint Quaternion Matrices

Zhe Ouyang1, Yun Wang*

College of Information Sciences and Engineering, Shandong Agricultural University, Tai’an Shandong

Received: May 5th, 2018; accepted: May 18th, 2018; published: May 25th, 2018

ABSTRACT

Quaternion matrix has a wide range of applications in the field of engineering technology, physics and computer science. In this paper, we describe the background and development of quaternion and quaternion matrices. Moreover some basic definitions and theorems of quaternion and quaternion matrices are demonstrated. Finally, we discuss the Jacobi iteration of right eigenvalues of real self-adjoint quaternion matrices based on the real-representation.

Keywords:Quaternion, Quaternion Matrices, Right Eigenvalue, Jacobi Iteration

Copyright © 2018 by authors and Hans Publishers Inc.

1. 引言

2. 基础知识

$q=a+bi+cj+dk\text{}a,b,c,d\in R$ (2.1.1)

$Q=\left\{a+bi+cj+dk|a,b,c,d\in R\right\}.$

${q}_{1}={a}_{1}+{b}_{1}i+{c}_{1}j+{d}_{1}k\in Q$${q}_{2}={a}_{2}+{b}_{2}i+{c}_{2}j+{d}_{2}k\in Q$ ，则两个四元数的相等、加法与乘法分别规定如下：

${q}_{1}={q}_{2}⇔{a}_{1}={a}_{2},{b}_{1}={b}_{2},{c}_{1}={c}_{2},{d}_{1}={d}_{2}$

${q}_{1}+{q}_{2}=\left({a}_{1}+{a}_{2}\right)+\left({b}_{1}+{b}_{2}\right)i+\left({c}_{1}+{c}_{2}\right)j+\left({d}_{1}+{d}_{2}\right)k$

$\begin{array}{c}{q}_{1}q{}_{2}=\left({a}_{1}{a}_{2}-{b}_{1}{b}_{2}-{c}_{1}{c}_{2}-{d}_{1}{d}_{2}\right)+\left({a}_{1}{b}_{2}+{b}_{1}{a}_{2}+{c}_{1}{d}_{2}-{d}_{1}{c}_{2}\right)i\\ +\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}+{b}_{2}{d}_{1}-{d}_{2}{b}_{1}\right)j+\left({a}_{1}{d}_{2}+{d}_{1}{a}_{2}+{b}_{1}{c}_{2}-{c}_{1}{b}_{1}\right)k\end{array}$

1) A的为复数的右特征值的集合 = ${A}_{\sigma }$ 的复特征值的集合；

2) A的右特征值的集合 = { ${a}^{-1}\lambda a$ | $0\ne a\in Q$$\lambda$${A}_{\sigma }$ 的复特征值}，

3. 自共轭实四元数矩阵的特征值

$A={A}_{0}+{A}_{1}i+A{}_{2}j+{A}_{3}k$

${A}^{\ast }=A$

${A}_{0}^{\text{T}}={A}_{0},\text{}{A}_{1}^{\text{T}}=-{A}_{1},\text{}{A}_{2}^{\text{T}}=-{A}_{2},\text{}{A}_{3}^{\text{T}}=-{A}_{3}$ (3.1.1)

$U+Vi+Wj+Gk$ ，其中 $U,V,W,G$ 均为实的n维列向量，则

$\left({A}_{0}+{A}_{1}i+{A}_{2}j+{A}_{3}k\right)\left(U+Vi+Wj+Gk\right)=\lambda \left(U+Vi+Wj+Gk\right)$ (3.1.2)

$\left(\begin{array}{cccc}{A}_{0}& -{A}_{1}& -{A}_{2}& -{A}_{3}\\ {A}_{1}& {A}_{0}& -{A}_{3}& {A}_{2}\\ {A}_{2}& {A}_{3}& {A}_{0}& -{A}_{1}\\ {A}_{3}& -{A}_{2}& {A}_{1}& {A}_{0}\end{array}\right)\left(\begin{array}{c}U\\ V\\ W\\ G\end{array}\right)=\lambda \left(\begin{array}{c}U\\ V\\ W\\ G\end{array}\right)$

$S=\left(\begin{array}{cccc}{A}_{0}& -{A}_{1}& -{A}_{2}& -{A}_{3}\\ {A}_{1}& {A}_{0}& -{A}_{3}& {A}_{2}\\ {A}_{2}& {A}_{3}& {A}_{0}& -{A}_{1}\\ {A}_{3}& -{A}_{2}& {A}_{1}& {A}_{0}\end{array}\right)$

${G}_{ij}\left(\theta \right)=\left(\begin{array}{ccccccccccc}1& & & & & & & & & & \\ & \ddots & & & & & & & & & \\ & & 1& & & & & & & & \\ & & & \mathrm{cos}\theta & & & & \mathrm{sin}\theta & & & \\ & & & & 1& & & & & & \\ & & & & & \ddots & & & & & \\ & & & & & & 1& & & & \\ & & & -\mathrm{sin}\theta & & & & \mathrm{cos}\theta & & & \\ & & & & & & & & 1& & \\ & & & & & & & & & \ddots & \\ & & & & & & & & & & 1\end{array}\right)$

${S}_{k}={G}_{k}{S}_{k-1}{G}_{k}^{\text{T}}\text{}\left(k=1,2,\cdot \cdot \cdot \right)$ (3.1.3)

4. 小结

The Jacobi Iteration of Eigenvalue of Real Self-Adjoint Quaternion Matrices[J]. 理论数学, 2018, 08(03): 273-277. https://doi.org/10.12677/PM.2018.83036

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