| 公理化的矩阵半张量积 |
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| 引用本文： | 程代展,冯俊娥.公理化的矩阵半张量积[J].控制理论与应用,2024,41(7):1172-1180. |
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| 作者姓名： | 程代展 冯俊娥 |
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| 作者单位： | 中国科学院数学与系统科学研究院,山东大学 |
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| 基金项目： | 国家自然科学基金项目(62350037, 62073315, 62273201, 12326425), 山东省泰山学者特聘教授基金项目(TSTP20221103)资助. |
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| 摘 要： | 本文给出矩阵半张量积的一个公理化框架, 它包括矩阵–矩阵半张量积、矩阵–向量半张量积和向量–向量半张量积. 首先, 对目前通用的各类矩阵半张量积的基本性质与应用做一个综述性的回顾. 然后, 介绍一种新近出现的矩阵半张量积, 即保维数矩阵半张量积. 跟普通矩阵乘法一样, 它是多功能的, 即它可同时实现矩阵–矩阵乘积、矩阵–向量乘积和向量–向量乘积这3种功能. 最后, 本文介绍保维数矩阵半张量积的一些代数性质, 包括非方矩阵的Cayley-Hamilton定理, 非方矩阵的特征值、特征向量等.
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| 关 键 词： | 公理化的矩阵半张量积 超矩阵 保维数矩阵半张量积 非方矩阵的Cayley-Hamilton定理 |
| 收稿时间： | 2023/7/24 0:00:00 |
| 修稿时间： | 2024/6/14 0:00:00 |
Axiomatized semi-tensor product of matrices |
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| CHENG Dai-zhan and Feng Jun-e.Axiomatized semi-tensor product of matrices[J].Control Theory & Applications,2024,41(7):1172-1180. |
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| Authors: | CHENG Dai-zhan and Feng Jun-e |
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| Affiliation: | Academy of Mathematics and Systems Science, Chinese Academy of Science,Shandong University |
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| Abstract: | An axiomatic framework is provided for the semi-tensor product (STP) of matrices, which consists of three kinds of STPs: matrix-matrix STP, matrix-vector STP and vector-vector STP. First, several commonly used STPs are surveyed with their basic properties and applications. Then, a newly appearing STP, called the dimension keeping STP, is introduced. The dimension keeping STP is a multi-functional STP, which can realize the functions of the matrix-matrix STP, matrix-vector STP, and vector-vector STP simultaneously. Finally, this paper also briefly introduces some algebraic properties brought by the dimension keeping STP, including Cayley-Hamilton theory of non-square matrices, eigenvalue and eigenvector of non-square matrices. |
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| Keywords: | axiomatized semi-tensor product hypermatrix dimension keeping semi-tensor product Cayley-Hamilton theory of non-square matrices |
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