{"title":"A new method based on the semi-tensor product of matrices for solving communicative quaternion matrix equation ∑i=1kAiXBi=C and its application","authors":"Mingcui Zhang, Ying Li, Jianhua Sun, Xueling Fan, Anli Wei","doi":"10.1016/j.bulsci.2025.103576","DOIUrl":null,"url":null,"abstract":"<div><div>This paper studies the least squares problem of the commutative quaternion matrix equation <span><span>(1.1)</span></span>, finds its minimal norm least squares (anti-)Hermitian solution. In the process of completing this work, we generalize the semi-tensor product of real matrices to the commutative quaternion matrices, then use it to extend the vector operators to the commutative quaternion matrix and propose the <em>L</em>-representation, which transforms the intricate commutative quaternion matrix equation into a solvable system of real linear equations, we also use <em>GH</em>-representation to reduce the complexity of the operation and greatly save the operation time. This can be illustrated by numerical examples in the paper. In addition, we take a special kind of commutative quaternion: reduced biquaternion as an example, and compare our method with another method in reference <span><span>[33]</span></span> to prove the effectiveness of our method. Finally, we apply the method used in this paper to symmetric color image restoration.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103576"},"PeriodicalIF":1.3000,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin des Sciences Mathematiques","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0007449725000028","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper studies the least squares problem of the commutative quaternion matrix equation (1.1), finds its minimal norm least squares (anti-)Hermitian solution. In the process of completing this work, we generalize the semi-tensor product of real matrices to the commutative quaternion matrices, then use it to extend the vector operators to the commutative quaternion matrix and propose the L-representation, which transforms the intricate commutative quaternion matrix equation into a solvable system of real linear equations, we also use GH-representation to reduce the complexity of the operation and greatly save the operation time. This can be illustrated by numerical examples in the paper. In addition, we take a special kind of commutative quaternion: reduced biquaternion as an example, and compare our method with another method in reference [33] to prove the effectiveness of our method. Finally, we apply the method used in this paper to symmetric color image restoration.