This paper addresses the problem of decomposing a given matrix into its sparse and low-rank components, which is NP-hard in general. The authors propose a convex optimization formulation using a combination of the $\ell_1$ norm and the nuclear norm to split the matrix into its components. They introduce the concept of *rank-sparsity incoherence*, an uncertainty principle between the sparsity pattern of a matrix and its row/column spaces, to characterize fundamental identifiability and sufficient conditions for exact recovery. The analysis is geometric, focusing on the tangent spaces to the algebraic varieties of sparse and low-rank matrices. The paper provides deterministic conditions for exact recovery based on the quantities $\xi(M)$ and $\mu(M)$, which measure the diffusion of elements in the tangent space and the spectrum of elements in the support, respectively. These conditions are satisfied with high probability when the sparse and low-rank matrices are drawn from certain random ensembles. The paper also discusses applications in graphical modeling, matrix rigidity, system identification, and optical systems. Finally, simulation results validate the theoretical findings.This paper addresses the problem of decomposing a given matrix into its sparse and low-rank components, which is NP-hard in general. The authors propose a convex optimization formulation using a combination of the $\ell_1$ norm and the nuclear norm to split the matrix into its components. They introduce the concept of *rank-sparsity incoherence*, an uncertainty principle between the sparsity pattern of a matrix and its row/column spaces, to characterize fundamental identifiability and sufficient conditions for exact recovery. The analysis is geometric, focusing on the tangent spaces to the algebraic varieties of sparse and low-rank matrices. The paper provides deterministic conditions for exact recovery based on the quantities $\xi(M)$ and $\mu(M)$, which measure the diffusion of elements in the tangent space and the spectrum of elements in the support, respectively. These conditions are satisfied with high probability when the sparse and low-rank matrices are drawn from certain random ensembles. The paper also discusses applications in graphical modeling, matrix rigidity, system identification, and optical systems. Finally, simulation results validate the theoretical findings.