# Lipschitz continuity of singular values

How smooth are the singular values of a matrix F in terms of entries of F? I am hoping for Lipschitz continuity, but was not able to find it.

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The singular values of $F$ are the (square roots of ) eigenvalues of $F F^t,$ and the regularity of the latter have been studied half-to-death. See either T. Kato (perturbation theory of linear operators, ch. 1) or Golub-van Loan (Matrix Computations -- they almost certainly talk about singular values directly, without going through eigenvalues, but at worst talk about eigenvalues).