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：1997年Kilmer与Martin[26]引入的张量t-积，是用于三阶张量分析与计算的有力工具。本文引入t-积下三阶张量的特征管与特征片，它们分别对应于矩阵特征值与特征向量的类比概念。我们研究了特征管与特征片的性质，并阐述了其数值计算方法，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表现。