Complex eigenvalue

Question

Question: Suppose is a 3×3 matrix with real entries that has a complex eigenvalue −1+8 with corresponding eigenvector

\begin{bmatrix} 1-2i\\ -1\\ 8i \end{bmatrix}

Find another eigenvalue and eigenvector for .

What should i do?

Answer 1

$A$ has real entries so $\bar{A}=A$. Now just conjugate $Av=\lambda v$ you get that $\bar{\lambda}$ is an eigenvalue corresponding to $\bar{v}$. (Remember this is only true because $A$ has real entries).

So $-1-8i$ is an eigenvalue with eigenvector $(1+2i,-1,-8i)^T$

Answer 2

Real entries means the characteristic polynomial is real, so another eigenvalue is...? Then think about $$\begin{bmatrix} a & b & c \\ * & * & * \\ * & * & * \\ \end{bmatrix}v=(-1+8i)v$$ might be tweaked when $(-1+8i)$ is replaced by the other eigenvalue. Brute force it if you must then look for the pattern.