 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/6/iii/solution

The matrix is

M = 2 I - e e^{T}

, where

e = (1, 1, 1)^{T}

. Along

\overset{e}{^} = e / 3

2 - 3 = - 1

e

has eigenvalue two. For example, use the orthonormal basis

u_{1} = \frac{( 1 , - 1 , 0 )}{2}, u_{2} = \frac{( 1 , 1 , - 2 )}{6}, u_{3} = \frac{( 1 , 1 , 1 )}{3} .

The principal-coordinate equation is

2 ξ^{2} + 2 η^{2} - ζ^{2} = 3, \frac{ξ ^{2} + η ^{2}}{3/2} - \frac{ζ ^{2}}{3} = 1.

It depends only on distance from the

u_{3}

axis and position along it, so every rotation about that axis preserves the surface. Therefore

the rotational axis is the line x_{1} = x_{2} = x_{3}, with direction (1, 1, 1) .

3/2

. The repeated positive eigenvalue makes the two transverse principal axes nonunique.

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