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.

翻译：2011 年, Kilmer 和 Martin 提出第三顺序的 强单值分解( T- SVD ) 。 自此以后, T- SVD 在高近近、 强光恢复、 多视图组群、 多视图特征提取、 高素描等应用了低级的 高压近似、 高光谱、 高光谱等应用。 通过 Discrete Fourier 变异、 矩阵 SVD 和 反 DFT, 第三个顺序的 高光谱被映射为 F- diagonal 第三顺序 。 我们称此为 Kilmer- Martin 映射为 。 我们显示, 如果第三顺序的 Exronor 正在以某些或多向的 高光标获取 T 产品, 则第三个顺序的 Kilmer- Martin 映射图是不可变的 。 我们根据 Kilmer- Martin 映射图, 和 最佳的 TRK- sir 的 10 讨论 。