Untitled

Abstract

This is a longer version of our article Burda et al., Phys. Rev. E82, 061114 (2010), containing more detailed explanations and providing pedagogical introductions to the methods we use. We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so-called M -transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.

Asymmetric correlation matrices: an analysis of financial data

“…the eigenvectors corresponding to the largest eigenvalues. In Figure 11 the components of V (1) are reported, dis- related to the largest eigenvalue λ 1 of the joint correlation matrix in equation (20). The distribution of components related to S&P stocks is plotted with the solid line, while the one of components related to FTSE stocks is plotted with the dashed line.…”

“…The eigenvalue spectrum of the joint correlation matrix in equation (20) displays one main bulk (see Figure 10), plus a few eigenvalues "leaking out" of such bulk. Some of those can already be seen in Figure 10, but not the largest two, equal to λ 1 = 112.7 and λ 2 = 31.8, i.e.…”

“…10. Eigenvalue spectrum of the joint correlation matrix in equation (20). For better visualization, the two largest eigenvalues, equal to 112.7 and 31.8, have not been plotted.…”

“…However, in [28] an equation for the average eigenvalue density for a product of an arbitrary number of large Gaussian random matrices was derived. Such equation was derived by means of a planar diagram expansion (see [29] for a step by step introduction to this technique) under the assumption of all matrix dimensions going to infinity with their ratios kept fixed. Also, quite importantly for our present discussion, the aforementioned equation can be solved exactly for the product of two matrices, as in equation (7).…”

“…Quite surprisingly, in those works most of the eigenvalue spectrum was shown to be fell fitted by a Marčenko-Pastur distribution, whereas only few, larger, eigenvalues were shown to carry relevant information on the market correlation structure by "leaking out" of the Marčenko-Pastur region. Ever since such works, physicists kept on analyzing financial correlation matrices, constantly refining the general picture described in [6,7] with increasing levels of insight [8,9,10,11,12,13,14,15,16,17,18,19], and also generalizing the framework defined by equation (2) to also include the effects due to temporal correlations [20,21].…”