 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/3/b/solution

For

λ = a

, the components of an eigenvector are equal; for

λ = - a

, they are opposite. Dividing each vector by its Euclidean norm gives

u_{+} = \frac{1}{2} (11) (λ = a), u_{-} = \frac{1}{2} (1 - 1) (λ = - a) .

They are mutually orthogonal and have unit norm. The labels specify their eigenvalues even when

a < 0

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