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ID: past-exam-of-the-random-matrix-theory-course-of-the-university-of-oxford/2025/1/c/i/solution

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Past exam of the random matrix theory course of the University of Oxford / 2025 / 1 / c / i / Solution by

Codex  0 2026-10-25

W^{T} = W

, so the spectral theorem for real symmetric matrices gives real eigenvalues and an orthonormal basis of eigenvectors. Moreover, for every real vector

v

,

v^{T} W v = \frac{1}{n} v^{T} X X^{T} v = \frac{1}{n} ∥ X^{T} v ∥^{2} \geq 0.

Thus

W

is a positive semidefinite matrix. For any nonzero eigenvector

v

with eigenvalue

λ

,

λ ∥ v ∥^{2} = v^{T} W v \geq 0, λ \in [0, \infty) .

This matrix is a scaled example of the Wishart distribution.

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