This paper presents a method for decomposing a matrix into its sparse and low-rank components using convex optimization. The matrix is formed by adding an unknown sparse matrix and an unknown low-rank matrix. The goal is to recover these components from the given matrix. The method uses a convex relaxation approach, minimizing a combination of the ℓ₁ norm (for sparsity) and the nuclear norm (for low-rankness). A key concept introduced is rank-sparsity incoherence, which measures the uncertainty principle between the sparsity pattern of a matrix and its row and column spaces. This concept is used to characterize the identifiability of the decomposition and to derive sufficient conditions for exact recovery. The analysis is geometric, involving tangent spaces to the algebraic varieties of sparse and low-rank matrices. The paper shows that when the sparse and low-rank matrices are drawn from certain natural random ensembles, the sufficient conditions for exact recovery are satisfied with high probability. The method is applied to various applications, including graphical modeling with latent variables, matrix rigidity, composite system identification, and partially coherent decomposition in optical systems. The paper also provides a detailed analysis of the optimality conditions for the convex program and proves that under certain conditions, the unique optimum of the program corresponds to the true sparse and low-rank components. The results are validated through simulation experiments on synthetic matrix decomposition problems, demonstrating the effectiveness of the method. The paper concludes with implications for the matrix rigidity problem and highlights the importance of the rank-sparsity incoherence in ensuring the success of the decomposition method.This paper presents a method for decomposing a matrix into its sparse and low-rank components using convex optimization. The matrix is formed by adding an unknown sparse matrix and an unknown low-rank matrix. The goal is to recover these components from the given matrix. The method uses a convex relaxation approach, minimizing a combination of the ℓ₁ norm (for sparsity) and the nuclear norm (for low-rankness). A key concept introduced is rank-sparsity incoherence, which measures the uncertainty principle between the sparsity pattern of a matrix and its row and column spaces. This concept is used to characterize the identifiability of the decomposition and to derive sufficient conditions for exact recovery. The analysis is geometric, involving tangent spaces to the algebraic varieties of sparse and low-rank matrices. The paper shows that when the sparse and low-rank matrices are drawn from certain natural random ensembles, the sufficient conditions for exact recovery are satisfied with high probability. The method is applied to various applications, including graphical modeling with latent variables, matrix rigidity, composite system identification, and partially coherent decomposition in optical systems. The paper also provides a detailed analysis of the optimality conditions for the convex program and proves that under certain conditions, the unique optimum of the program corresponds to the true sparse and low-rank components. The results are validated through simulation experiments on synthetic matrix decomposition problems, demonstrating the effectiveness of the method. The paper concludes with implications for the matrix rigidity problem and highlights the importance of the rank-sparsity incoherence in ensuring the success of the decomposition method.