Eigenvalue statistics of random real matrices

Completing Ginibre's work we determine the joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations. The symmetry parameter {τ} may vary from -1 (antisymmetric ensemble) through 0 (completely asymmetric ensemble) to +1 (symmetric ensemble). The elliptic law for the average density of eigenvalues in the limit of large dimension is recovered. Matrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.