 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/14/a/iv/solution

For the particular eigenvectors chosen in (iii), the dot products are

e_{1} \cdot e_{2} = 0, e_{1} \cdot e_{3} = 2, e_{2} \cdot e_{3} = - 6.

Thus only $e_{1}$ and $e_{2}$ are orthogonal for this choice. The matrix is not symmetric, so distinct eigenvalues alone do not imply orthogonality. Another basis within the repeated eigenspace may change which chosen vectors are perpendicular; the dot products above specify the convention completely.

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