# Further results on the Drazin inverse of even-order tensors

## TL;DR

This paper advances the theory of the Drazin inverse for even-order tensors, providing new characterizations, computational methods, and applications to solving multilinear systems with convergence analysis.

## Contribution

It introduces new characterizations of the Drazin inverse and W-weighted Drazin inverse for tensors, along with methods for computation and solving tensor systems.

## Key findings

- New characterizations of tensor Drazin inverse
- Methods for computing the inverse using generalized inverses
- Convergence analysis of iterative solution methods

## Abstract

The notion of the Drazin inverse of an even-order tensor with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402-3413]. In this article, we further elaborate this theory by producing a few characterizations of the Drazin inverse and the W-weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution to the multilinear systems using the Drazin inverse and iterative (higher order Gauss-Seidel) method of tensors. Besides this, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.