For a linear operator $T:V\to V$ over $\mathbb F$, a scalar $\lambda\in\mathbb F$ is an eigenvalue if some nonzero $v$ satisfies $T(v)=\lambda v$. The same operator may have different eigenvalue availability over different fields. A $90^\circ$ real planar rotation has no real eigenvalues but has complex eigenvalues $\pm i$ after complexification.
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