Can a real eigenvalue (of a matrix that has two real and two complex conjugate eigenvalues) correspond to a complex eigenvector? Thanks in advance.
2026-04-06 02:47:41.1775443661
Real eigenvalues with complex eigenvectors?
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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 $1$.