Let $S$ be a symmetrix $n \times n$ matrix. (Remember that means $S^t = S$). Let $v$, $w$ be eigenvectors of $S$ for eigenvalues $\lambda$, $\mu$ respectively. Suppose $\lambda \not= \mu$. Show that $v \cdot w = 0$.
Hints give:
- In other words, the eigenspaces of symmetric matrices are perpendicular to each other.
- Use a: Where you let $x, y \in \mathbb{R}^n$. Verify that $(Sx) \cdot y = x \cdot (Sy)$. Answer: I verified that they do equal.