Eigenvalues of $AA^T$ if $A^T A=I_r$

Question

Suppose $A$ is a $n \times r$ matrix with $n>r$ and $A^TA=I_r$, i.e., $A$ could be the matrix of eigenvectors of $r$ eigenvalues. I am wondering what's the eigenvalues of $AA^T$, which is of rank $r$ and sum of all eigenvalues of $AA^T$ equals $r$. Any particular formula for the eigenvalues of $AA^T$? Thanks!

Answer 1

Hint: for any $\lambda$ such that $A^T A x = \lambda x$, you have

$$ A^T A x = \lambda x \implies AA^T (Ax) = \lambda (Ax). $$