Singular values are not the magnitudes of eigenvalues

A week ago someone asked whether the singular values of a general (real) matrix are the magnitudes of its eigenvalues. There are various ways to see that the answer is no, but here’s an amusingly nonconstructive proof:

Consider a random matrix $A$ taken from $GL(n)$ with some smooth distribution. With probability 1 all singular values of $A$ will be unique. However, with nonzero probability $A$ will be near a rotation matrix and will have a complex conjugate pair of eigenvalues with the same magnitude. Therefore, the singular values of $A$ are not always the magnitudes of the eigenvalues.

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