Confusion about properties of orthogonal eigenvectors

Question

I'm trying to show that for any matrix $A$ $\in$ $\mathbb R$^m×n, if $v$₁ and $v$₂ are orthogonal eigenvectors of $A^TA$ then Av₁ and Av₂ are orthogonal.

I know that $A^TA$ is symmetric which means its eigenvectors are orthogonal and I have found that if $v$ is an eigenvector of $A^TA$ with eigenvalue $\lambda \neq 0$, then $Av$ is an eigenvector of $AA^T$. I feel that these properties are important in proving the above but I am unsure how.

Answer 1

$\langle Av_1 , Av_2 \rangle =\langle A^{T}Av_1 , v_2 \rangle =\lambda_1 \langle v_1, v_2 \rangle=0$ where $\lambda_1$ is the eigen value corresponding to $v_1$.