I'm having trouble with this problem, I'm lost as to how to approach it.
Let $A$ be an $m\times n$ matrix, and suppose $\mathbf{v}$ and $\mathbf{w}$ are orthogonal eigenvectors of $A^TA$. Show that $A\mathbf{v}$ and $A\mathbf{w}$ are orthogonal.
I can prove that the transpose of $A$ has the same eigenvalues as $A$, but I'm unsure how that might be helpful as they can have different eigenvectors. Any advice?