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) 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 applied 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 proposed 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 established 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.

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

“…The concepts of T-range space, T-null space, tensor norm, and T-Moore-Penrose inverse are given as follows [30].…”

“…The T-null space of A, N (A) := Null((F H p ⊗I n 1 )bcirc(A)(F p ⊗I n 1 )), 'Null' represents the null space, (3) The tensor unitary invariant norm A := bcirc(A) , where the matrix norm · should also be chosen as a unitary invariant norm. (4) The T-Moore-Penrose inverse [30] A † = bcirc −1 ((bcirc(A)) † ).…”

Matrix functions in network analysis

“…Several additional applications of generalized matrix functions (under the name of "singular functional calculus") can be found in [5]. For an extension to tensors, see [113].…”

Computation of outer inverses of tensors using the QR decomposition

“…Well-known results on the numerical methods for computing outer inverses are surveyed in the monographs (Wang et al 2018;Wei et al 2018). Additional results about the perturbation bounds for the Moore-Penrose inverse, Core, Core-EP, CMP, and the DMP inverses of tensors via Einstein product can be found in references (Ding and Wei 2016;Jin et al 2017;Ma 2018;Ma et al 2019;Miao et al 2020;Wang et al 2020;Du et al 2019;Wang et al 2020;Xie et al 2019;Wang and Wei 2017).…”