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

First $W^T=W$, so the spectral theorem for real symmetric matrices gives real eigenvalues and an orthonormal basis of eigenvectors. Moreover, for every real vector $v$,Thus $W$ is a positive semidefinite matrix. For any nonzero eigenvector $v$ with eigenvalue $\lambda$,This matrix is a scaled example of the Wishart distribution.