This paper presents a method for exactly recovering low-rank matrices from a subset of their entries using convex optimization. The key idea is to minimize the nuclear norm of a matrix, which is the sum of its singular values, subject to the constraint that the observed entries match those of the original matrix. The authors show that if the number of observed entries is sufficiently large, then this optimization problem can perfectly recover the original matrix. They derive conditions on the number of observed entries required for this recovery, showing that it is possible to recover most low-rank matrices from a small number of randomly sampled entries. The results are connected to the field of compressed sensing, where similar techniques are used to reconstruct signals from incomplete measurements. The paper also discusses the implications of these results for various applications, including recommender systems and sensor network localization. The authors conclude that their method provides a tractable and efficient way to recover low-rank matrices from incomplete data.This paper presents a method for exactly recovering low-rank matrices from a subset of their entries using convex optimization. The key idea is to minimize the nuclear norm of a matrix, which is the sum of its singular values, subject to the constraint that the observed entries match those of the original matrix. The authors show that if the number of observed entries is sufficiently large, then this optimization problem can perfectly recover the original matrix. They derive conditions on the number of observed entries required for this recovery, showing that it is possible to recover most low-rank matrices from a small number of randomly sampled entries. The results are connected to the field of compressed sensing, where similar techniques are used to reconstruct signals from incomplete measurements. The paper also discusses the implications of these results for various applications, including recommender systems and sensor network localization. The authors conclude that their method provides a tractable and efficient way to recover low-rank matrices from incomplete data.