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