A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions.

In this paper, we propose an efficient numerical algorithm for solving large-scale ill-posed linear inverse problems encountered in image restoration. To boost computational efficiency, we extend the structured fast iterative shrinkage-thresholding algorithm (sFISTA) for addressing the corresponding $l_1$-regularized minimization problem, and further introduce the over-relaxation technique to accelerate the algorithm. The proposed algorithm is termed structured over-relaxed monotone FISTA (sOMFISTA). The convergence analysis of sOMFISTA is also conducted. The algorithmic framework of sOMFISTA is universally applicable to any non-smooth convex regularization term, exhibiting remarkable flexibility. Extensive numerical experiments are carried out to systematically validate the superiority in efficiency and performance of the proposed sOMFISTA.