# Real eigenvalues with complex eigenvectors?

Can a real eigenvalue (of a matrix that has two real and two complex conjugate eigenvalues) correspond to a complex eigenvector? Thanks in advance.

• Sure, and you don't even need the complex eigenvalues. E.g., $[i , 0]$ is an eigenvector of the identity matrix. – eyeballfrog Apr 27 at 17:54
• @eyeballfrog though the identity matrix does not have necessarily complex eigenvectors – Henry Apr 27 at 17:56

Yes. Take, for instance, the matrix$$\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{bmatrix}.$$Its eigenvalues are $$1$$ (multiplicity two), $$i$$, and $$-i$$. And $$(i,i,0,0)$$ is an eigenvector corresponding to the eigenvalue $$1$$.