Do real matrices always have real eigenvalues? 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?

No, a real matrix does not necessarily have real eigenvalues; an example is $\pmatrix{0&1\\-1&0}$.
On the other hand, since this matrix happens to be orthogonal and has the eigenvalues $\pm i$ -- for eigenvectors $(1\mp i, 1\pm i)$ -- I think you're supposed to consider only real eigenvalues in the first place.
