Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 13 / a / ii

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- Solution ii

Solution

For either sign, the dot product with

v_{1}

is

v_{1} \cdot v_{\pm} = (- 5 \pm 15) + (2 \mp 15) + 3 = 0.

The remaining pair also satisfies

v_{+} \cdot v_{-} = (25 - 15) + (4 - 15) + 1 = 0 .

Hence the three eigenvectors are mutually orthogonal. This also follows from the spectral theorem for real symmetric matrices, since

M = M^{T}

and its eigenvalues are distinct, but the displayed products verify the actual chosen vectors directly.

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