Eigenvalues that are complex numbers Have a square matrix problem that involves complex numbers and am at a loss.  $M$ is a square matrix with real entries.  $\lambda = a + ib$ is a complex eigenvalue of $M$, show that the complex conjugate $\bar{\lambda} = a - ib$ of $\lambda$ is also an eigenvalue of $M$.  Does solving this relate to a matrix having a characteristic polynomial of $(t^2-4)$? Could the community please explain? 

let the eigenvector corresponding to the eigenvalue $a+ib$ be $u+iv$ with both $u$ and $v$ real and $|u|^2 + |v|^2 \neq 0$. then we have $M(u+iv) = (a+ib)(u+iv)$ can be separated to give $Mu = au - bv, Mv = av + bu.$  
now, $M(u -iv) = Mu - iMv = (au-bv) -i(av+bu) = a(u-iv) -ib(u-iv) = (a-ib)(u-iv)$ this shows that $a-ib$ is an eigenvalue of $M.$
