Property: For a symmetric matrix $A$, eigenvectors belong to different eigenspaces are mutually orthogonal.

Proof:

Suppose we have

$Ax_1=\lambda_1 x_1$
$Ax_2=\lambda_2 x_2$

and $\lambda_1\ne\lambda_2$.

Then $x_2^T Ax_1=x_2^T \lambda_1 x_1$. Note $x_2^T Ax_1=(Ax_2)^T x_1=\lambda_2x_2^T x_1$. So

$\lambda_2 x_2^Tx_1=\lambda_1 x2^T x_1$.

Because $\lambda_1\ne\lambda_2$, so $x_2^Tx_1=0$ which implies $x_2 \perp x_1$.

Remark: for complex and self-adjoint matrix, the property still hold.