Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 15C / d

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- Solution d

Solution

The three eigenvalues are

1, e^{iπ /4} = \frac{1 + i}{2}, e^{3 iπ /4} = \frac{- 1 + i}{2} .

Thus

N (U)

is the filled triangle with vertices

(1, 0), (2^{- 1/2}, 2^{- 1/2}), (- 2^{- 1/2}, 2^{- 1/2})

in the complex plane. Its upper edge is the horizontal chord joining the last two vertices, and the other two edges join those points to

1

. In particular, the origin lies outside this triangle.

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