Matrix and its inverse have the same singular values

Question

What is known about matrices which have the same singular values as their inverses? Note that this is equivalent to saying that if $A$ satisfies this condition and $\sigma$ is a singular value of $A$, then $1/\sigma$ is also a singular value of $A$.

Answer 1

The singular values of an invertible (real) matrix $A$ are the square roots of the eigenvalues of $A^T A$ (which are the same as those of $A A^T$). If $B = A^T A$, you're saying that $B$ and $B^{-1}$ have the same eigenvalues.