Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 18 / b / iv

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- Solution iv

Solution

Choose the unnormalized eigenvectors

u_{+} = (1, 1)^{T}, S u_{+} = 3 u_{+}, u_{-} = (1, - 1)^{T}, S u_{-} = - u_{-} .

Their linear combination giving the specified vector is

x = \frac{1}{2} u_{+} + \frac{1}{2} u_{-} .

Applying the reciprocal eigenvalues of the inverse matrix therefore yields

y = \frac{1}{6} u_{+} - \frac{1}{2} u_{-} .

In Cartesian coordinates, this is

(- 1/3, 2/3)^{T}

, and direct matrix multiplication gives

S y = (1, 0)^{T}

.

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