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. 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 .
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 .
R
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
It was shown recently that the f-diagonal tensor in the
R
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
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