Low-rank matrix recovery (LR) model, aiming at decomposing a matrix into a low-rank matrix and a sparse matrix, has shown the potential to address the problem of salient object detection, where the decomposed low-rank matrix represents image background, and the sparse matrix identifies salient objects. However, there still exist two deficiencies. First, since the model assumes the elements in the sparse matrix are mutually independent, it ignores the spatial and pattern relation of image regions. Second, when the low-rank and sparse matrices are coherent, such as the salient objects and background share similar appearance or the background is complex, it is difficult for LR model to separate them. To address these problems, we propose a novel structured matrix decomposition (SMD) model. In the model, a tree-structured sparsity-inducing regularization is firstly introduced to capture the structure of image and enforce patches within the objects to have similar saliency values. Next, a Laplacian regularization is further imposed to enlarge the difference between the representations of salient objects and background. Finally, high-level priors are integrated to guide the matrix decomposition and boost the detection. We evaluate our model for salient object detection on 5 datasets, namely MSRA10K, DUT-OMRON, iCoSeg, SOD and ECSSD, in terms of single object, multiple objects and complex scene images. We show competitive results as compared with 24 state-of-the-art methods with respect to 7 metrics.