Before revision: One of the early ideas started from the 2004 workshop is to regard third order tensors as tubal matrices such as matrix theory can be naturally extended to third order tensors. We revisit this approach in this paper. We define the value of a tubal scalar to remedy the gap between this approach and the practical needs to have singular values as real scalars. We define the transpose of a tubal scalar such that the transpose of a tubal matrix is a natural extension of the transpose of a matrix and still corresponding to the transpose of a third order tensor. 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.
After revision: It was shown recently that the f-diagonal tensor in the T-SVD factorizations.