A note on eigenvalue of tensors and its application

The tensor eigenvalue problem has been widely studied in recent years. In this paper, several new properties of eigenvalues and determinants of tensors are explored. We also proposed a formula to compute the determinant of a tensor as a mimic of the matrix determinant. The Perron-Frobenius theorem, one of the most important results in non-negative matrix theory, is proposed for the class of non-negative tensors in the Einstein product framework. Further, the power method, a widely used matrix iterative method for finding the largest eigenvalue, is framed for tensors using the Einstein product. The proposed higher-order power method is applied to calculate the largest eigenvalue of the Laplacian tensors associated with hyper-stars and hyper-trees. The numerical results show that the higher-order power method with the Einstein product is stable.