  We investigate the level-density $\sigma(x)$ and level-spacing distribution
$p(s)$ of random matrices $M=AF\neq M^{\dagger}$ where $F$ is a (diagonal)
inner-product and $A$ is a random, real symmetric or complex Hermitian matrix
with independent entries drawn from a probability distribution $q(x)$ with zero
mean and finite higher moments. Although not Hermitian, the matrix $M$ is
self-adjoint with respect to $F$ and thus has purely real eigenvalues. We find
that the level density $\sigma_F(x)$ is independent of the underlying
distribution $q(x)$, is solely characterized by $F$, and therefore generalizes
Wigner's semicircle distribution $\sigma_W(x)$. We find that the level-spacing
distributions $p(s)$ are independent of $q(x)$, are dependent upon the
inner-product $F$ and whether $A$ is real or complex, and therefore generalize
the Wigner's surmise for level spacing. Our results suggest $F$-dependent
generalizations of the well-known Gaussian Orthogonal Ensemble (GOE) and
Gaussian Unitary Ensemble (GUE) classes.
