Orthogonal Eigenvectors

Question

Let $v_1$ and $v_2$ be eigenvectors of $A^{T}A$ . Prove if $v_1$ and $v_2$ are orthogonal, then $Av_1$ and $Av_2$ are also orthogonal.

Beginning Proof:Let $v_1$ and $v_2$ be eigenvectors of $A^{T}A$. By definition, we can write $A^{T}Av_{1}=\lambda v_1$ (equ. 1) and $A^{T}Av_{2}=\lambda v_2$ (equ. 2). We want to show that $Av_1 \cdot Av_2=0$

After this, I am lost. I want to do some manipulation with Equ. 1 and 2, but I am not sure exactly how helpful that will be.

Answer 1

$v_1\perp v_2$ means $v_1\cdot v_2=v_1^Tv_2=0$, so

$$Av_1\cdot Av_2=v_1^TA^T Av_2=\lambda_2v_1^Tv_2=0$$