If $A$ is $n\times n$, the singular values of $A$ are the squares of the eigenvalues of $A^{T}A$. But how does one obtain the eigenvalues of A given the singular values of A (A is unknown)?
EDIT: One way that just popped into my mind would be to use SVD: Multiply the identity matrix (which is an orthonormal basis U) by $\Sigma$ (diagonal matrix with singular values) and then multiply again by the identity(which is an orthonormal basis $V^T$) - basically just ending up with $\Sigma$. Afterwards, simply computing the characteristic polynomial of $\Sigma$, then solving for eigenvalues.