| 张量与张量转置的特征值问题 |
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| 引用本文： | 陈震,卢琳璋,刘智秉.张量与张量转置的特征值问题[J].厦门大学学报(自然科学版),2012,51(1):9-11. |
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| 作者姓名： | 陈震 卢琳璋 刘智秉 |
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| 作者单位： | 1. 贵州师范大学数学与计算机科学学院,贵州贵阳550001;厦门大学数学科学学院,福建厦门361005
2. 九江学院理学院,江西九江,332005 |
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| 基金项目： | 国家自然科学基金项目，贵州省科学技术基金项目 |
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| 摘 要： | 研究了张量与张量转置的特征值问题,证明了张量的mode-p,特征对与张量p-转置的mode-i特征对是相同的,通过例子说明了张量与张量p-转置的mode-i特征值不一定相等,当mode-i特征值μ1和mode-j特征值μ1不相等时,对应的特征向量也不一定正交.从而说明了矩阵特征值的性质并不能完全推广到张量情形.此外,还给出了张量与张量p-转置的mode-i特征对相等时指标向量p需要满足的条件.
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| 关 键 词： | 张量 张量转置 特征值 特征向量 正交 |
The Eigenvalue Problems for Tensor and Tensor Transposition |
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| CHEN Zhen
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LU Lin-zhang
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LIU Zhi-bing.The Eigenvalue Problems for Tensor and Tensor Transposition[J].Journal of Xiamen University(Natural Science),2012,51(1):9-11. |
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| Authors: | CHEN Zhen LU Lin-zhang LIU Zhi-bing |
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| Institution: | 1.School of Mathematics and Computer Science,Guizhou Normal University,Guiyang 550001,China;2.School of Mathematical Sciences,Xiamen University,Xiamen 361005,China;3.School of Science,Jiujiang University,Jiujiang 332005,China) |
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| Abstract: | The tensor eigenproblem has many important applications.This paper studies the eigenproblem for tensor and tensor transposition,and proves that the mode-pi eigenpair of tensor is the same as the mode-i eigenpair of p-transposition of tensor.It is illustrated by examples that the mode-i eigenvalue of tensor is not necessarily equal to that of p-transposition of tensor,and the eigenvectors corresponding to the mode-i eigenvalue μi and the mode-j eigenvalue μj are not orthogonal when μi≠μj.Therefore,the results show that the properties of eigenvalues of matrix can not be completely extended to tensor.Based on this conclusion,this paper gives a condition that the index vector p should satisfy when the mode-i eigenpair of tensor is equal to that of the p-transposition of tensor. |
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| Keywords: | tensor tensor transposition eigenvalue eigenvector orthogonality |
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