The Eigenvalue Problems for Tensor and Tensor Transposition

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.