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Matrices are a fundamental mathematical object that plays a fundamental role in our description of nature. They are often costly to compute element-by-element. This paper proposes a two-pronged method for computing matrices less expensively: first, cheaply identifying a basis in which the matrix is sparse; and second, applying the ideas of compressed sensing to recover this sparse matrix at a low cost which scales only with the number of non-zero entries in the matrix. After developing the method and giving numerical examples, we apply it to the problem of recovering the vibrational modes and frequencies of a molecule. A basis in which the quantum mechanical Hessian is sparse is obtained via a low-cost molecular mechanics calculation, and compressed sensing is then used to obtain the full quantum mechanical Hessian at approximately one-third the typical computational cost.