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? 

If $Mv=\lambda v$ where $v=(v_1,...,v_n)$ is a vector of complex numbers, then $$M\bar v=\overline{Mv}=\overline{\lambda v}=\bar\lambda \bar v$$
Here by writing a line over a matrix (or vector) I mean to take the complex conjugate of every entry; the first equality holds because $M$ is real, so $\bar M=M$, and because as you can check directly $\bar M\bar v=\overline{Mv}$ for any complex matrix $M$ and vector $v$. So $\bar v$ is a $\bar\lambda$-eigenvector of $M$.
This does relate to the characteristic polynomial $\chi M$: every eigenvalue is a root of $\chi M$, and if $M$ is real then $\chi M$ has real coefficients, and if a real polynomial $p$ has a complex root $\lambda$ then $\bar \lambda$ is a root as well. 
