It was shown recently that the f-diagonal tensor in the T-SVD factorization must satisfy some special properties. Such f-diagonal tensors are called s-diagonal tensors. In this paper, we show that such a discussion can be extended to any real invertible linear transformation. We show that two Eckart-Young like theorems hold for a third order real tensor, under any doubly real-preserving unitary transformation. The Discrete Fourier Transformation matrix and any orthogonal matrices are doubly real-preserving unitary transformations. In particular, Discrete Cosine Transformation is in this category. We use tubal matrices as a tool for our study. We feel that the tubal matrix language makes this approach more natural.