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In this paper, we are concerned with a class of tensor equations. Some explicit solutions to these equations are obtained. They are extensions of some results given by Lancaster. At the end of the article, these results are also employed to prove Lyapunov and Stein stability theorems for tensors. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solution to the least squares problem for the quaternion Sylvester tensor equation. An iterative algorithm based on tensor format is presented to solve this problem. The convergence properties of the proposed iterative method are studied. We also consider the best approximate problem related to quaternion Sylvester tensor equation. Numerical examples are provided to confirm the theoretical results, which demonstrate that the proposed algorithm is effective and feasible for solving the least squares problem of the quaternion Sylvester tensor equation. [ABSTRACT FROM AUTHOR]

We propose a class of iterative algorithms to solve some tensor equations via Einstein product. These algorithms use tensor computations with no matricizations involved. For any (special) initial tensor, a solution (the minimal Frobenius norm solution) of related problems can be obtained within finite iteration steps in the absence of roundoff errors. Numerical examples are provided to confirm the theoretical results, which demonstrate that this kind of iterative methods are effective and feasible for solving some tensor equations. [ABSTRACT FROM AUTHOR]