Block-Diagonalization of Quaternion Circulant Matrices with Applications

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-08-01 DOI:10.1137/23m1552115

Junjun Pan, Michael K. Ng

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1429-1454, September 2024.
Abstract. It is well known that a complex circulant matrix can be diagonalized by a discrete Fourier matrix with imaginary unit [math]. The main aim of this paper is to demonstrate that a quaternion circulant matrix cannot be diagonalized by a discrete quaternion Fourier matrix with three imaginary units [math], [math], and [math]. Instead, a quaternion circulant matrix can be block-diagonalized into 1-by-1 block and 2-by-2 block matrices by permuted discrete quaternion Fourier transform matrix. With such a block-diagonalized form, the inverse of a quaternion circulant matrix can be determined efficiently similarly to the inverse of a complex circulant matrix. We make use of this block-diagonalized form to study quaternion tensor singular value decomposition of quaternion tensors where the entries are quaternion numbers. The applications, including computing the inverse of a quaternion circulant matrix and solving quaternion Toeplitz systems arising from linear prediction of quaternion signals, are employed to validate the efficiency of our proposed block- diagonalized results.

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四元圆周矩阵的对角分块及其应用

SIAM 矩阵分析与应用期刊》，第 45 卷第 3 期，第 1429-1454 页，2024 年 9 月。摘要众所周知，复圆周矩阵可以用带虚单元的离散傅里叶矩阵对角化[math]。本文的主要目的是证明一个四元环矩阵不能被一个具有三个虚数单位[math]、[math]和[math]的离散四元傅里叶矩阵对角化。相反，四元环矩阵可以通过包络离散四元傅里叶变换矩阵分块对角化为 1-by-1 分块矩阵和 2-by-2 分块矩阵。有了这种分块对角化形式，四元环矩阵的逆就能像复数环矩阵的逆一样有效地确定。我们利用这种分块对角化形式来研究四元张量的奇异值分解，其中的条目是四元数。我们利用计算四元环形矩阵的逆和求解四元信号线性预测中产生的四元托普利兹系统等应用来验证我们提出的分块对角化结果的效率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.