Linear Algebra › Section E · Eigenvalues and eigenvectors › Topic E.5
Complex eigenvalues
A rotation turns every vector off its line, so it has no real eigenvectors. Allowing complex numbers recovers the eigenvalues, and their size and angle reveal a hidden rotation-and-scaling inside any real 2 × 2 matrix of this kind.
In this topic
- When real eigenvectors run out
- Complex vectors and the arithmetic you need
- Eigenvectors of the rotation
- Conjugate pairs
- Rotation and scaling in one matrix
- Worked example: finding a complex eigenpair
- The real factorisation
- Worked example: rotation hidden in a matrix
- What repeated multiplication does
- Common mistakes to avoid
- Key takeaways
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