Tubal Matrix Analysis
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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 study on tubal matrices may reveal more links between matrix theory and tensor theory.
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