Before revision:  We define positive semi-definite symmetric tubal scalars such that s-diagonal tubal matrices can be defined meaningfully and are still corresponding to third order s-diagonal tensors, which were introduced recently for studying the T-SVD factorizations. We show that the 2-norm of a tubal matrix is equal to its largest T-singular value, multiplied with a coefficient, which is 1 in the case of matrices. This shows that T-singular values of tubal matrices are natural extensions of singular values of matrices. Further studyon tubal matrices may reveal more links between matrix theory and tensor theory .
After revision:  We define positive semi-definite symmetric tubal scalars such that s-diagonal tubal matrices can be defined meaningfully and are still corresponding to third order s-diagonal tensors, which were introduced recently for studying 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 .