Given a set $S$ of complex numbers such that $z\in S \implies \bar{z}\in S$ can I find a real matrix $M$ (in a space of dimension $|S|$) whose eigenvalues are precisely those in $S$? More generally is there a way to know when a real matrix M does exist?
I'm struggling to come up with any counter examples. I thought about proving the stronger statement "Every complex matrix is similar to a real matrix" but I don't think thats easier to show. Something like jordan normal form could be useful but we don't really care about the jordan structure, just the eigenvalues. Another stronger (but less so than the last) that could work is "Every diagonalisable complex matrix is similar to a real matrix". There are also some nice ideas like knowing that the determinant and trace of a real matrix are real but the conjugate property probably makes this useless.
Best Answer
For the case $S=\{a+ib,a-ib\}$ we have the matrix $\begin{pmatrix}a & b\\-b &a\end{pmatrix}$.
For larger sets $S$ just take a matrix with appropriate $2\times 2$ blocks down the diagonal.
For your subsidiary questions clearly $\begin{pmatrix}i\end{pmatrix}$ is not similar to a real matrix.
[Note: Some people even define complex numbers to be these $2\times 2$ matrices.]