Can I find eigenvalues for $A$ if I know the eigenvalues from $AA^T$

Question

Let's say we have a real matrix $A$ and I find the eigenvalues for $\sqrt{AA^T}$. Is it possible for me then to find the eigenvalues for $A$ without using

$$det(A-\lambda I)=0$$

?

Answer 1

No, take $$ A=\begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}.$$ The eigenvalues of $A^TA$ are $1$ with geometric multiplicity two. If we knew that $A$ were positive (i.e. self-adjoint and having only positive eigenvalues), then taking square roots would recover the eigenvalues. Basically, the issue here is the sign.