This paper presents the first theoretical analysis of the performance of alternating minimization for low-rank matrix completion and matrix sensing. Alternating minimization is a widely applicable and empirically successful approach 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 alternating between finding the best $U$ and the best $V$. Each sub-problem is typically convex and tractable, but the overall problem is non-convex. The paper shows that under certain conditions, similar to those used by existing methods, alternating minimization guarantees faster convergence to the true matrix compared to existing methods. Specifically, for matrix sensing, the algorithm achieves geometric convergence in $O(\log(1/\epsilon))$ steps, while for matrix completion, it requires $O(\log(\|M\|_F / \epsilon))$ steps. The analysis also addresses the issue of the number of measurements required, showing that the algorithm can recover the true matrix with a near-optimal number of measurements. The paper provides a detailed analysis for both rank-1 and rank-$k$ matrices, demonstrating the effectiveness of the alternating minimization approach in practical scenarios.This paper presents the first theoretical analysis of the performance of alternating minimization for low-rank matrix completion and matrix sensing. Alternating minimization is a widely applicable and empirically successful approach 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 alternating between finding the best $U$ and the best $V$. Each sub-problem is typically convex and tractable, but the overall problem is non-convex. The paper shows that under certain conditions, similar to those used by existing methods, alternating minimization guarantees faster convergence to the true matrix compared to existing methods. Specifically, for matrix sensing, the algorithm achieves geometric convergence in $O(\log(1/\epsilon))$ steps, while for matrix completion, it requires $O(\log(\|M\|_F / \epsilon))$ steps. The analysis also addresses the issue of the number of measurements required, showing that the algorithm can recover the true matrix with a near-optimal number of measurements. The paper provides a detailed analysis for both rank-1 and rank-$k$ matrices, demonstrating the effectiveness of the alternating minimization approach in practical scenarios.