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I have a matrix that gives singular values which are quite small. I would like to perturb it in a way that keeps it close to the original matrix, only it has larger singular values. For a square matrix, a way to do it would be to add $\lambda I$ to the matrix. Is there a way to do it for a non-square matrix?

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Why not just compute $U\Sigma V^*$, change the chosen diagonal elements of $\Sigma$ to $\Sigma'$, and reconstruct the perturbation as $U\Sigma' V^*$? – whuber♦ Jan 24 '12 at 19:16
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Maybe you can provide us a little detail on the intended application. It could be that solving the underlying problem may provide an alternative to what you state here. – cardinal Jan 24 '12 at 20:58

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