This paper presents the first theoretical analysis of the performance of alternating minimization for low-rank matrix completion and matrix sensing. The alternating minimization approach represents a widely used and empirically successful method for finding low-rank matrices that best fit given data. The method involves representing the target matrix in a bi-linear form $ X = UV^\dagger $, and then alternately optimizing for $ U $ and $ V $. While each sub-problem is convex and tractable, the overall problem is non-convex, and there has been limited theoretical understanding of when this approach yields good results. The paper shows that alternating minimization succeeds under similar conditions to those used by existing methods for matrix completion and matrix sensing. For matrix sensing, the method is shown to converge geometrically to the true matrix under the restricted isometry property (RIP) condition. For matrix completion, the method is shown to recover the true matrix under incoherence conditions. The paper also shows that alternating minimization guarantees faster convergence to the true matrix compared to existing results, while allowing a simpler analysis. The paper presents theoretical results for both matrix sensing and matrix completion. For matrix sensing, the method is shown to converge geometrically to the true matrix under the RIP condition. For matrix completion, the method is shown to recover the true matrix under incoherence conditions. The paper also shows that alternating minimization requires only a small number of computationally cheap iterations, making it more efficient than existing methods. The paper also presents a stagewise version of the alternating minimization algorithm for matrix sensing, which reduces the number of measurements required. The stagewise algorithm is shown to achieve near-optimal measurement requirements. The paper also presents a stagewise version of the alternating minimization algorithm for matrix completion, which is shown to achieve near-optimal measurement requirements. The paper concludes that alternating minimization is a promising approach for low-rank matrix completion and matrix sensing, and that it has the potential to be more efficient than existing methods. The paper also shows that alternating minimization has the potential to be more accurate than existing methods.This paper presents the first theoretical analysis of the performance of alternating minimization for low-rank matrix completion and matrix sensing. The alternating minimization approach represents a widely used and empirically successful method for finding low-rank matrices that best fit given data. The method involves representing the target matrix in a bi-linear form $ X = UV^\dagger $, and then alternately optimizing for $ U $ and $ V $. While each sub-problem is convex and tractable, the overall problem is non-convex, and there has been limited theoretical understanding of when this approach yields good results. The paper shows that alternating minimization succeeds under similar conditions to those used by existing methods for matrix completion and matrix sensing. For matrix sensing, the method is shown to converge geometrically to the true matrix under the restricted isometry property (RIP) condition. For matrix completion, the method is shown to recover the true matrix under incoherence conditions. The paper also shows that alternating minimization guarantees faster convergence to the true matrix compared to existing results, while allowing a simpler analysis. The paper presents theoretical results for both matrix sensing and matrix completion. For matrix sensing, the method is shown to converge geometrically to the true matrix under the RIP condition. For matrix completion, the method is shown to recover the true matrix under incoherence conditions. The paper also shows that alternating minimization requires only a small number of computationally cheap iterations, making it more efficient than existing methods. The paper also presents a stagewise version of the alternating minimization algorithm for matrix sensing, which reduces the number of measurements required. The stagewise algorithm is shown to achieve near-optimal measurement requirements. The paper also presents a stagewise version of the alternating minimization algorithm for matrix completion, which is shown to achieve near-optimal measurement requirements. The paper concludes that alternating minimization is a promising approach for low-rank matrix completion and matrix sensing, and that it has the potential to be more efficient than existing methods. The paper also shows that alternating minimization has the potential to be more accurate than existing methods.