A study on the 1-$\Gamma$ inverse of tensors via the M-Product

In this paper, we will study the issue about the 1-$\Gamma$ inverse, where $\Gamma\in\{†, D, *\}$, via the M-product. The aim of the current study is threefold. Firstly, the definition and characteristic of the 1-$\Gamma$ inverse is introduced. Equivalent conditions for a tensor to be a 1-$\Gamma$ inverse are established. Secondly, using the singular value decomposition, the corresponding numerical algorithms for computing the 1-$\Gamma$ inverse are given. Finally, the solutions of the multilinear equations related 1-$\Gamma$ inverse are studied, and numerical calculations are given to verify our conclusions.