 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015/6/solution

(5 - λ) (3 - λ) - 3 = (λ - 6) (λ - 2)

. Corresponding normalized eigenvectors are

λ = 6 : e_{6} = \frac{1}{2} (3, 1), λ = 2 : e_{2} = \frac{1}{2} (- 1, 3) .

These form an orthonormal basis. Write

x = u e_{6} + v e_{2}

6 u^{2} + 2 v^{2} = 4, \frac{u ^{2}}{2/3} + \frac{v ^{2}}{2} = 1.

The locus is an ellipse centred at the origin. Its minor semiaxis has length

2/3

along the direction

3 0^{\circ}

from the positive horizontal axis; its major semiaxis has length

2

along

12 0^{\circ}

. The right panel of the figure in part 1 shows these principal directions.

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