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This ability of compressed sensing to recover a sparse matrix with a  number of measurements that scales just linearly with the non-zero elements of the matrix opens new possibilities for the calculations of matrices, even if a basis where the basis is sparse is not known.  The idea of finding a basis where the matrix is approximately diagonal is the basis of perturbation theory, for example. However in general this methods involve approximations with an error that depends on how good the eigenvalues are approximated.   By using compressed sensing, however, the approximated basis does not influence the error in the result. Only the computational cost.  begs the obvious question: for a given scientific application, how can  we find a basis in which the matrix we wish to calculate is sparse?  Of course, every diagonalizable matrix has a sparse basis, namely the