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