Exercise 8

Suppose that $Q$ is an orthogonal matrix.

1. Remembering that $\vvec\cdot\wvec=\vvec^T\wvec$, explain why $$Q\xvec\cdot(Q\yvec) = \xvec\cdot\yvec.$$

2. Explain why $\len{Q\xvec} = \len{\xvec}$.

   This means that the length of a vector is unchanged after multiplying by an orthogonal matrix.

3. If $\lambda$ is a real eigenvalue of $Q$, explain why $\lambda=\pm1$.