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Weighted nuclear norm minimization has been recently recognized as a technique for reconstruction of a low-rank matrix from compressively sampled measurements when some prior information about the column and row subspaces of the matrix is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted nuclear norm minimization when multiple weights are allowed. This setup might be used when one has access to prior subspaces forming multiple angles with the column and row subspaces of the ground-truth matrix. While existing works in this field use a single weight to penalize all the angles, we propose a multi-weight problem which is designed to penalize each angle independently using a distinct weight. Specifically, we prove that our proposed multi-weight problem is stable and robust under weaker conditions for the measurement operator than the analogous conditions for single-weight scenario and standard nuclear norm minimization. Moreover, it provides better reconstruction error than the state of the art methods. We illustrate our results with extensive numerical experiments that demonstrate the advantages of allowing multiple weights in the recovery procedure.