Eigenvalues of a matrix of complex numbers

Question

Can a matrix of complex numbers have real eigenvalues ? Imaginary part of the complex number are not equal to zero.

I know Hermitian matrix can have real eigenvalues, but what about non-Hermitian ones?

Answer 1

Take for instance the matrix $$ m =\begin{bmatrix} 1&0\\ 0&2 \end{bmatrix} $$ and conjugate it with a complex matrix; for instance, conjugate $m$ with $$ t = \begin{bmatrix} 1+i & i\\2+i & i\end{bmatrix} $$ to get $$ t^{-1} m t = \begin{bmatrix}3+i & i \\ -3+i & -i \end{bmatrix}, $$ which has still eigenvalues $1$ and $2$.