Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix?

Question

Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix? I believe they do not since, for instance, I was taught that the trace is the sum of the eigenvalues and this would mean that this is not always the case. What puzzles me is that I have seen people claiming that some real matrix has no eigenvalues simply because the eigenvalues were all complex numbers. Were they right?

Answer 1

That depends upon the context. If, say, you have a $n\times n$ real matrix and and someone asks you what are its eigenvalues, then it is implicit that you should provide the real eigenvalues. And perhaps that there is none. On the other hand, it may be implicit that, in fact, you are after the complex eigenvalues.

On the other hand, if the matrix is real, then the trace is real, but it may well have non-real eigenvalues. For instance, $\left[\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right]$ has null trace, but its eigenvalues are $\pm i$.

Answer 2

A real matrix can have no eigenvalues in exactly the same way that the real polynomial equation $x^2+1=0$ can have no solutions.

Whether it is true or not depends entirely on who is taking, who they're talking to, and what they are talking about.