Eigenvalues of a Random Matrix

If $X$ is a random matrix, the eigenvectors and eigenvalues of $X$ are random as well. Thus the vector $v$ and the real number $\lambda$ in your example are random.

Edit: At the risk of belaboring the obvious, in this context, the random matrix $X$ is a function $X:\Omega\to\mathcal M_n(\mathbb R)$ and an eigenvector and an eigenvalue of $X$ are functions $v:\Omega\to\mathbb R^n\setminus\{0\}$ and $\lambda:\Omega\to\mathbb C$ such that, for every $\omega$ in $\Omega$, $X(\omega)v(\omega)=\lambda(\omega)v(\omega)$. For example, the first coordinate of the vector $X(\omega)v(\omega)$ is $\lambda(\omega)v_1(\omega)$.

I think for every "point" in your sample space you get a matrix- and hence can look at the eigenvalues of that matrix. Thus we get the "eigenvalue distribution", i.e. it becomes a random variable in its own right.