= Solution
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> $v$,
$$
v^TWv=\frac1n v^TXX^Tv=\frac1n\lVert X^Tv\rVert^2\geq0.
$$
Thus $W$ is a <positive semidefinite matrix>. For any nonzero <eigenvector> $v$ with <eigenvalue> $\lambda$,
$$
\lambda\lVert v\rVert^2=v^TWv\geq0,
\qquad\boxed{\lambda\in[0,\infty).}
$$
This <matrix> is a scaled example of the <Wishart distribution>.
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