The most classical problem in random matrix theory is to specify a natural joint distribution for the entries of a large random matrix, then study the asymptotic behavior of the distribution of the eigenvalues. I will describe joint work with Elizabeth Meckes on the opposite problem: For a natural model of random matrices with prescribed eigenvalues, we study the asymptotic behavior of the distribution of the matrix entries. Our results have applications to quantum mechanics, and shed new light on the universality phenomenon in classical random matrix theory.