The Equivalence Canonical Forms of Two Sets of Five Quaternion Matrices With Applications

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2024-12-30 DOI:10.1002/mma.10613

Zhuo-Heng He, Yun-Ze Xu, Qing-Wen Wang, Chong-Quan Zhang

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引用次数: 0

Abstract

In this paper, the equivalence canonical forms of two sets of five quaternion matrices are considered. The simultaneous decompositions of these two sets of quaternion matrices are established. Some practical algorithms for computing the decompositions are also presented. Using the equivalence canonical forms, some practical solvability conditions for two systems of generalized Sylvester-type quaternion matrix equations are derived in terms of the ranks of the coefficient matrices. The general solutions to the systems are also provided. Some numerical examples are given to illustrate the results. The proposed simultaneous decompositions are applied to construct a new framework of color watermarks embedding. In this framework, five color watermarks can be embedded into one host image simultaneously with high efficiency. Moreover, a counterexample for the existence of the equivalence canonical form of the general matrix quaternity with the same row or column numbers is presented. The results extend the existing findings on the equivalence canonical forms of general matrix arrays.

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两个五四元数矩阵集的等价标准形式及其应用

本文研究了两组五四元数矩阵的等价标准形式。建立了这两组四元数矩阵的同时分解。并给出了一些实用的分解算法。利用等价标准形式，导出了两类广义sylvester型四元数矩阵方程的实用可解条件，即系数矩阵的秩。并给出了系统的一般解决方案。给出了一些数值算例来说明结果。采用该方法构建了一种新的彩色水印嵌入框架。该框架可以高效地同时嵌入5个彩色水印到一幅主图像中。此外，还给出了具有相同行数或列数的一般矩阵四元群等价标准形式存在的一个反例。所得结果推广了已有的关于一般矩阵阵列等价规范形式的研究结果。

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

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.