A new preconditioned Gauss-Seidel method for solving M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document}-tensor multi-linear system

By utilizing some elements of each row of the majorization matrix associated with the coefficient tensor, we propose a preconditioner, and present the corresponding preconditioned Gauss–Seidel method for solving M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document}-tensor multi-linear system. Theoretically, we give the convergence and comparison theorems of the proposed preconditioned Gauss–Seidel method. Numerically, we show the correctness of theoretical results and the efficiency of the proposed preconditioner by some examples.