发表机构
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences; Research Center of Semi-tensor Product of Matrices-Theory and Applications, Liaocheng; Dept. of Mathematics, Shandong University; State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences; School of Mathematical Sciences, Nanjing Normal University（中国科学院数学与系统科学研究院系统与控制重点实验室; 聊城半张量积矩阵理论及应用研究中心; 山东大学数学系; 中国科学院数学与系统科学研究院数学科学国家重点实验室; 中国科学院大学数学科学学院; 南京师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将保维半张量积下的非方矩阵环推广到超矩阵，提出四类超矩阵特征值/特征向量，结合非方束克罗内克标准形与首一分解算法，构造出兼具矩阵若尔当标准形性质的张量标准形，可直接计算超矩阵的所有特征值/特征向量。
AI 中文摘要
研究基于保维（DK-）半张量积（STP）的非方矩阵环，将该环结构推广到超矩阵，针对超矩阵预先指定的特殊矩阵化方式，分别提出四类超矩阵的特征值/特征向量（EEs）：普通EE（OEE）、通用EE（UEE）、对角EE（DEE）、水平DEE（HDEE）。利用非方束的克罗内克标准形（KCF）计算OEE，再用首一分解算法（MDA）分别提取UEE、DEE和MEE，最后通过非方束的KCF构造张量的KCF，该KCF可揭示张量的所有EEs，且不仅具有矩阵若尔当标准形的类似性质，还将其作为特例包含，超矩阵的所有EEs均可通过其KCF直接计算。
英文摘要
The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.
