Matrix, singular values vs Eigen values

Question

I know that the singular values of matrix A are the root for the eigenvalues for $A^TA$ But is that also true for $AA^T$? and when I should use the first and when the latter if both give the same result?

Answer 1

If $A\in\mathbb{R}^{m\times n}$, then you should use $A^TA$ when $m>n$ and you should use $AA^T$ when $m<n$. The reason for this is that the number of singular values is necessarily equal to $\min\{m,n\}$ from the definition of the SVD. One should then use the corresponding "second-order" matrix that has $\min\{m,n\}$ eigenvalues, as the other will have the same eigenvalues with an additional $\max\{m-n,n-m\}$ eigenvalues equal to zero since $\mathrm{rank}(A)\leq\min\{m,n\}$.