Before revision:  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.
After revision:  Here, an s-diagonal tensor is a special 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.