Practice question · Multiple choice
A rotation of the plane by 90° has no real eigenvectors. Why should that be obvious before any computation?
Hints
- Draw any line through the origin and rotate it 90°. Does it land on itself?
- Ask what an eigenvector is required to do under the transformation.
Show the answer
B. Because a 90° rotation moves every direction off its own line
Why
Every line is moved to a perpendicular one, so no direction survives, and the characteristic polynomial λ² + 1 duly has no real roots. It is also why complex eigenvalues are not a technicality: they encode rotation, which is exactly the behaviour that has no invariant real direction.
Practise Eigenvalues and Eigenvectors
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