The tensor t-product, introduced by Kilmer and Martin [26], is a powerful tool for the analysis of and computation with third-order tensors. This paper introduces eigentubes and eigenslices of third-order tensors under the t-product. The eigentubes and eigenslices are analogues of eigenvalues and eigenvectors for matrices. Properties of eigentubes and eigenslices are investigated and numerical methods for their computation are described. The methods include the tensor power method, tensor subspace iteration, and the tensor QR algorithm. Computed examples illustrate the performance of these methods.

翻译：由Kilmer和Martin[26]提出的张量t-积是三阶张量分析与计算的有力工具。本文引入了三阶张量在t-积下的特征管和特征切片。特征管和特征切片对应于矩阵特征值与特征向量的类比概念。研究了特征管与特征切片的性质，并描述了其数值计算方法。该方法包括张量幂法、张量子空间迭代法以及张量QR算法。计算实例展示了这些方法的性能。