Consider random matrices $X,Y\in \mathbb{R}^{N\times P}$, i.e., $X_{ik}\sim{\cal N}(0,1/N)$ i.i.d. and likewise for $Y$.
- The eigenvalues of the symmetric covariance matrix $X^TX$ reside on the real line, follow the Marchenko-Pastur distribution, and have support at $[\lambda_-, \lambda_+]$ for $\lambda_\pm=(1\pm\sqrt{\alpha})^2, \alpha=P/N$.
- The eigenvalues of the asymmetric matrix $X^TY$ are complex and follow the Circular law for a circle of radius $\sqrt{\alpha}$ around the origin of the complex plane.
Can you say anything exact about the eigenvalues distribution of their (negative) sum: $$X^TY-X^TX$$ If those $X^TX, X^TY$ were free (in the sense of Free Probability Theory), the distribution of the sum would have been a convolution of the two eigenvalue distributions and we could have calculated a formula for it in terms of their R-Transform. However, if I understand correctly, those are not free because of the common $X$ term.