# Eigenvalues of a matrix of complex numbers

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?

• if this weren't possible, quantum mechanics would be in trouble! (in other words, see the spectral theorem.) – symplectomorphic May 1 '14 at 17:30

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 $2$.