Past exam of the random matrix theory course of the University of Oxford / 2025 / 3 / a / i

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- Solution i

Solution

The characteristic polynomial of the rotation matrix is

det (M - z I) = (cos U - z)^{2} + sin^{2} U = z^{2} - 2 z cos U + 1.

Its roots are

cos U \pm i sin U

, so

e^{i Θ^{+}} = e^{i U}, e^{i Θ^{-}} = e^{- i U}, Θ^{\pm} = \pm U (mod 2 π) .

For example

(1, i)^{T}

is an eigenvector for

e^{i U}

and

(1, - i)^{T}

for

e^{- i U}

. At

U = 0

or

π

the two eigenvalues coincide; these events have probability zero under the uniform distribution.

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