A large number of problems arising in computer vision can be reduced to the problem of minimizing the nuclear norm of a matrix, subject to additional structural and sparsity constraints on its elements. Examples of relevant applications include, among others, robust tracking in the presence of outliers, manifold embedding, event detection, inpainting and tracklet matching across occlusion. In principle, these problems can be reduced to a convex semi-definite optimization form and solved using interior point methods. However, the poor scaling properties of these methods limit the use of this approach to relatively small sized problems. The main result of this paper shows that structured nuclear norm minimization problems can be efficiently solved by using an iterative Augmented Lagrangian Type (ALM) method that only requires performing at each iteration a combination of matrix thresholding and matrix inversion steps. As we illustrate in the paper with several examples, the proposed algorithm results in a substantial reduction of computational time and memory requirements when compared against interior-point methods, opening up the possibility of solving realistic, large sized problems.