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The tensor t-product, introduced by Kilmer and Martin [26], is a powerfultool for the analysis of and computation with third-order tensors. This paperintroduces eigentubes and eigenslices of third-order tensors under thet-product. The eigentubes and eigenslices are analogues of eigenvalues andeigenvectors for matrices. Properties of eigentubes and eigenslices areinvestigated and numerical methods for their computation are described. Themethods include the tensor power method, tensor subspace iteration, and thetensor QR algorithm. Computed examples illustrate the performance of thesemethods.

Spectral computation with third-order tensors using the t-product