Generalized Tensor Function via the Tensor Singular Value Decomposition based on the T-Product

Abstract

In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) via the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors via the T-product, from which the projection operators and Moore Penrose inverse of tensors are also obtained. We also establish the Cauchy integral formula for tensors by using the partial isometry tensors and applied it into the solution of tensor equations. In the last part of our paper, we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are also listed.