  In 2011, Kilmer and Martin proposed tensor singular value decomposition
(T-SVD) for third order tensors. Since then, T-SVD has applications in low rank
tensor approximation, tensor recovery, multi-view clustering, multi-view
feature extraction, tensor sketching, etc. By going through the Discrete
Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is
mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin
mapping. We show that the Kilmer-Martin mapping of a third order tensor is
invariant if that third order tensor is taking T-product with some orthogonal
tensors. We define singular values and T-rank of that third order tensor based
upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular
values and T-singular values of a third order tensor are invariant when it is
taking T-product with some orthogonal tensors. Some properties of singular
values, T-rank and best T-rank one approximation are discussed.

Numerical Analysis;Numerical Analysis