﻿ 四元数矩阵特征值的Jacobi迭代

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

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 qua-ternion matrices are demonstrated. Finally, we discuss the Jacobi iteration of right eigenvalues of real self-adjoint quaternion matrices based on the real-representation.

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

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https://doi.org/10.1016/0024-3795(95)00543-9

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https://doi.org/10.1016/S0024-3795(00)00154-3

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https://doi.org/10.1016/S0024-3795(02)00276-8

[8] 黄敬频, 陆云双. 自共轭四元数循环矩阵的特征值问题[J]. 数学的实践与认识, 2016(13): 251-257.

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