Tubal Matrices Liqun Qi∗ and ZiyanLuo† June 15, 2021 Abstract 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 theo- rems hold for a third order real tensor, under any doubly real-preserving unitary transformation. The normalized Discrete Fourier Transformation (DFT) matrix, an arbitrary orthogonal matrix, the product of the normalized DFT matrix and an arbitrary orthogonal matrix are examples of doubly real-preserving unitary transformations. We use tubal matrices as a tool for our study. We feel that the tubal matrix language makes this approach more natural. Key words. Tubal matrix, tensor, T-SVD factorization, tubal rank, B-rank, Eckart-Young like theorems AMS subject classifications. 15A69, 15A18 1 Introduction arXiv:2105.00793v3 [math.NA] 14 Jun 2021 Tensor decompositions have wide applications in engineering and data science [11]. The most popular tensor decompositions include CP decomposition and Tucker decompo- sition as well as tensor train decomposition [11, 3, 17]. The tensor-tensor product (t-product) approach, developed by Kilmer, Martin, Bra- man and others [10, 1, 9, 8], is somewhat different. They defined T-product opera- tion such that a third order tensor can be regarded as a linear operator applied on ∗Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China; (
[email protected]). †Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China.