Spectral domain of large nonsymmetric correlated Wishart matrices.

We study complex eigenvalues of the Wishart model for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as C=AB(t)/T, where B(t) is the transpose of B and both matrices A and B are of dimensions N×T. If A and B are uncorrelated, or equivalently if C vanishes on average, it is known that at large matrix dimension the domain of the eigenvalues of C is a circle centered-at-origin and the eigenvalue density depends only on the radial distances. We consider actual correlation in A and B and derive a result for the contour describing the domain of the bulk of the eigenvalues of C in the limit of large N and T where the ratio N/T is finite. In particular, we show that the eigenvalue domain is sensitive to the correlations. For example, when C is diagonal on average with the same element c≠0, the contour is no longer a circle centered at origin but a shifted ellipse. In this case we explicitly derive a result for the spectral density which again depends only on the radial distances. For more general cases, we show that the contour depends on the symmetric and antisymmetric parts of the correlation matrix resulting from the ensemble-averaged C. If the correlation matrix is normal, then the contour depends only on its spectrum. We also provide numerics to justify our analytics.