I was trying to show that orthogonal matrices have eigenvalues 1 or -1.
Let $u$ be an eigenvector of $A$ (orthogonal) corresponding to eigenvalue $\lambda$. Since orthogonal matrices preserve length, $ \|Au\|=|\lambda|\cdot\|u\|=\|u\|$. Since $\|u\|\ne0$, $|\lambda|=1$.
Now I am stuck to show that lambda is only a real number. Can any one help with this?