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?

I  guess  it  depends  whether  you  are  working  with  vector  spaces  over  the  real  numbers  or  vector  spaces over  the  complex  numbers.In  the  latter case  the  answer  is  no, however  in  the  former  the  answer has  to  be  yes.Is  it  not  guys ?
