Spectral Domain of Large Nonsymmetric Correlated Wishart Matrices

Abstract

We study {the} complex eigenvalues of the Wishart model defined for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as \(\mathsf{C}=\mathsf{AB}^{t}/T\), where \(\mathsf{B}^{t}\) is the transpose of \(\mathsf{B}\) and both matrices \(\mathsf{A}\) and \(\mathsf{B}\) are of dimension \(N\times T\). We consider {\it actual} correlations between the matrices so that on the ensemble average \(\mathsf{C}\) does not vanish. We derive a loop equation for the spectral density of \(\mathsf{C}\) in {the} large \(N\) and \(T\) limit where the ratio \(N/T\) is finite. The actual correlations changes the complex eigenvalues of \(\mathsf{C}\), and hence their domain from the results known for the vanishing \(\mathsf{C}\) or for the uncorrelated \(\mathsf{A}\) and \(\mathsf{B}\). Using the loop equation we derive {a} result for the contour describing the domain of {the} bulk of the eigenvalues of \(\mathsf{C}\). If the nonvanishing-correlation matrix is diagonal with the same element \(c\ne0\), the contour is no longer a circle centered at origin but a shifted ellipse. In this case, the loop equation is analytically solvable and we explicitly derive {a} result for the spectral density. For more general cases, our analytical result implies that the contour depends on its symmetric and anti-symmetric parts if the nonvanishing-correlation matrix is nonsymmetric. On the other hand, if it is symmetric then the contour depends only on the spectrum of the correlation matrix. We also provide numerics to justify our analytics.