We show that the T-SVD factorization of a third order real tensor  by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT , decomposes that third order tensor to the T-product of two orthogonal tensors and an s-diagonal tensor . Here, an s-diagonal tensor is a special f-diagonal tensor. We call such a T-SVD factorization a standard T-SVD (ST-SVD) factorization. We show that an f-diagonal tensor is an s-diagonal tensor if and only if ST-TVD factorization diagonalizes it to itself . Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode is 2, 3 and 4, we present direct sufficient and necessary conditions which do not involve such a complex number.
A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT . We call such an f-diagonal tensor an s-diagonal tensor . An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values . Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode is 2, 3 and 4, we present direct sufficient and necessary conditions which do not involve such a complex number.
R
We show that the T-SVD factorization of a
A
A
is mapped to a special f-diagonal tensor
R
, decomposes that third order tensor to the T-product of two orthogonal tensors and an s-diagonal tensor . Here,
. We call such an f-diagonal tensor
R
is a special
. An
D
tensor. We call such a T-SVD factorization a standard T-SVD (ST-SVD) factorization. We show that an f-diagonal
R
ST-TVD factorization diagonalizes it to itself
it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values