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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 matrixwith independent entries drawn from a probability distribution $q(x)$ with zeromean and finite higher moments. Although not Hermitian, the matrix $M$ isself-adjoint with respect to $F$ and thus has purely real eigenvalues. We findthat the level density $\sigma_F(x)$ is independent of the underlyingdistribution $q(x)$, is solely characterized by $F$, and therefore generalizesWigner's semicircle distribution $\sigma_W(x)$. We find that the level-spacingdistributions $p(s)$ are independent of $q(x)$, are dependent upon theinner-product $F$ and whether $A$ is real or complex, and therefore generalizethe Wigner's surmise for level spacing. Our results suggest $F$-dependentgeneralizations of the well-known Gaussian Orthogonal Ensemble (GOE) andGaussian Unitary Ensemble (GUE) classes.

Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices