This paper introduces Robust Principal Component Analysis (RPCA), a method to recover a low-rank matrix and a sparse matrix from a data matrix that is the sum of the two. The method uses convex optimization, specifically Principal Component Pursuit (PCP), which minimizes a weighted combination of the nuclear norm (for the low-rank component) and the ℓ₁ norm (for the sparse component). The paper proves that under certain conditions, PCP can exactly recover both components even if a significant fraction of the data is corrupted. It also extends this approach to handle missing data and discusses applications in video surveillance and face recognition. The methodology is robust to outliers and can handle large-scale data efficiently. The paper also provides theoretical guarantees for the recovery of low-rank and sparse components under specific assumptions, including incoherence conditions and sparsity constraints. The results show that PCP can be applied to a wide range of real-world problems where data may be corrupted or missing.This paper introduces Robust Principal Component Analysis (RPCA), a method to recover a low-rank matrix and a sparse matrix from a data matrix that is the sum of the two. The method uses convex optimization, specifically Principal Component Pursuit (PCP), which minimizes a weighted combination of the nuclear norm (for the low-rank component) and the ℓ₁ norm (for the sparse component). The paper proves that under certain conditions, PCP can exactly recover both components even if a significant fraction of the data is corrupted. It also extends this approach to handle missing data and discusses applications in video surveillance and face recognition. The methodology is robust to outliers and can handle large-scale data efficiently. The paper also provides theoretical guarantees for the recovery of low-rank and sparse components under specific assumptions, including incoherence conditions and sparsity constraints. The results show that PCP can be applied to a wide range of real-world problems where data may be corrupted or missing.