The generalized inverses of tensors via the C-Product

TL;DR

This paper introduces new definitions and algorithms for generalized inverses of tensors under the C-Product, including Moore-Penrose, Drazin, and inverse along a tensor, with various decomposition-based expressions.

Contribution

It defines and develops algorithms for generalized inverses of tensors under the C-Product, expanding tensor inverse theory and computational methods.

Findings

Defined Moore-Penrose, Drazin, and inverse along tensors under C-Product

Provided decomposition-based expressions for these inverses

Established algorithms for computing these tensor inverses

Abstract

This paper studies the issues about the generalized inverses of tensors under the C-Product. The aim of this paper is threefold. Firstly, this paper present the definition of the Moore-Penrose inverse, Drazin inverse of tensors under the C-Product. Moreover, the inverse along a tensor is also introduced. Secondly, this paper gives some other expressions of the generalized inverses of tensors by using several decomposition forms of tensors. Finally, the algorithms for the Moore-Penrose inverse, Drazin inverse of tensors and the inverse along a tensor are established.