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Exercise 8.3.2

Avg difficulty 3.0

If $A$ is the matrix of a linear transformation that rotates all vectors in $\mathbb{R}^{2}$ by $60^{\circ}$ , explain why $A$ cannot have any real eigenvalues. Is there an angle such that rotation by this angle would have a real eigenvalue? What eigenvalues would be obtainable in this way?

Selinger, Matrix Theory and Linear Algebra, Eigenvalues, eigenvectors, and diagonalization, Geometric interpretation of eigenvectors, Exercise 8.3.2

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.