Complex eigenvectors with conjugate entries

Question

Consider the matrix $$A = \begin{bmatrix}2&-10\\1&4\end{bmatrix}$$ The eigenvalues of this matrix are complex: $\lambda_{1,2} = 3 \pm 3i$. Now I calculated the eigenvectors to be $ \begin{bmatrix}10\\-1\mp3i\end{bmatrix}$respectively for each eigenvalue. So my first though is: The complex entries are conjugates. My question: Is that always so (I suspect YES, and that would make life a lot easier) and if so, is there a proof? (This is part of my learning process)

Answer 1

If the matrix $A$ is real then the eigenvalues will occur in conjugate pairs.

If $Av = \lambda v$, then $A \bar{v} = \bar{\lambda} \bar{v}$.