Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 3

Table of contents
- a 3
  - Solution a
- b 3
  - Solution b

a

Solution

The characteristic polynomial is

det (- λ a a - λ) = λ^{2} - a^{2} .

Its roots give the eigenvalues

λ = a, λ = - a .

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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