Showing two vectors are orthogonal

Question

Let $A$ be a symmetric $n \times n$ real matrix, where $n \geq 4$.

Let $v_1, \dots, v_4$ be nonzero vectors.

Suppose $A v_i = (2i-1) v_i$.

Why must $(v_1 + 2 v_2 ) \cdot (3 v_3 + 4v_4 ) = 0$ ?

Answer 1

Hint

Eigen-vectors of distinct eigen-values are mutually perpendicular.