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

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