Are there any results which discuss when non-zero singular values $\sigma_i$ of a square (complex) matrix $A$ are all the same?
This question could be reframed as when are the non-zero eigenvalues of $AA^{\dagger}$ the same, but I couldn't seem to find much on this either.
An alternative, tangential, question could also be, given that square matrix $B$ has all non-zero singular values equal, what properties does square matrix $C$ have such that the singular values of $CB$ are still all equal (although as I understand it, this is perhaps a harder problem).
Any avenues of discussion or useful references would be greatly appreciated!