TITLE: Does $A$ and $A^TA$ have the same eigenvalues
QUESTION [5 upvotes]: Do matrices $A$ and $A^TA$ have the same eigenvalues?
I think they do but I can't find anything online about it

REPLY [3 votes]: If $\lambda$ is an eigenvalue of $\bf A$ with eigenvector $\bf x$, we have  $\mathbf{Ax}=\lambda\bf x$
Then, $(\mathbf{A^\top A})\mathbf x=\mathbf A^\top(\lambda \mathbf x)=(\lambda \mathbf A^\top)\bf x$
If $\bf A$ is symmetric, the above reduces to $(\mathbf{A^\top A})\mathbf x=\lambda^2\bf x$
So your claim is only true when $\mathbf A=\mathbf I$ $\quad($which implies $\lambda=1)$.