Generalized tensor function via the tensor singular value decomposition based on the T-product

• Published in: Linear algebra and its applications Vol. 590; pp. 258 - 303
• Main Authors: Miao, Yun, Qi, Liqun, Wei, Yimin
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.04.2020; American Elsevier Company, Inc
• Subjects: Block tensor multiplication; Cauchy integral formula; Complex-to-real isomorphism; Cones; Decomposition; Generalized tensor function; Invariants; Isomorphism; Jordan algebra; Lie algebra; Linear algebra; Mathematical analysis; Moore-Penrose inverse; Operators (mathematics); Singular value decomposition; T-CSVD; T-product; T-SVD; Taylor series; Tensor bilinear form; Tensor-to-matrix isomorphism; Tensors; T-product; Complex-to-real isomorphism; Tensor bilinear form; T-CSVD; 15A48; 15A69; Generalized tensor function; Moore-Penrose inverse; Jordan algebra; 65F10; Cauchy integral formula; Tensor-to-matrix isomorphism; 65H10; Block tensor multiplication; Lie algebra; 65N22; T-SVD
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2019.12.035

In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore-Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and apply it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and propose theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator establishes an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions. The concept of invariant tensor cones is raised.