  Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an
$(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly
at random from the matrix analogue to the classical $\ell_ p^n$-ball, defined
as the set of all self-adjoint $(n\times n)$-matrices satisfying $\sum_{k=1}^n
|\lambda_k(A)|^p\leq 1$. We prove a large deviations principle for the (random)
spectral measure of the matrix $n^{1/p} Z_n$. As a consequence, we obtain that
the spectral measure of $n^{1/p} Z_n$ converges weakly almost surely to a
non-random limiting measure given by the Ullman distribution, as $n\to\infty$.
The corresponding results for random matrices in Schatten trace classes, where
eigenvalues are replaced by the singular values, are also presented.
