 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/5/b/solution

λ^{2} - 2 λ cos θ + 1

cos θ \pm i sin θ

. A real diagonal form can have only real eigenvalues, so it requires

sin θ = 0

. Conversely, when

θ = kπ

, the rotation matrix is already

(- 1)^{k} I

. Thus

real diagonalization is possible exactly when θ \in π Z .

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