The 2×2 identity matrix has every unit vector as an eigenvector, but most symmetric matrices near the identity have just two orthogonal unit vectors (and their opposites) as eigenvectors. They could in fact be any unit vectors: show that if θ is given, there exists a symmetric matrix S such that I + εS has eigenvectors (cos θ, sin θ) and (sinθ,−cosθ) for any ε ̸= 0.
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