Numerical ranges of non-normal random matrices: elliptic Ginibre and non-Hermitian Wishart ensembles

Abstract

The numerical range of a non-normal matrix plays a central role as a descriptor of non-normal effects beyond spectral information. We study a class of fundamental non-Hermitian random matrix ensembles that interpolate between the Hermitian and non-Hermitian regimes. Our analysis focuses on the elliptic Ginibre ensemble and its chiral counterpart, as well as on non-Hermitian Wishart matrices. For each of these models, we explicitly characterise the geometry of the numerical range in the large-system limit. In particular, we show that for the elliptic Ginibre ensemble and its chiral version, the limiting numerical range is an ellipse, whereas for the non-Hermitian Wishart ensemble it is described by a non-elliptic envelope.

Figures (4)

• Figure 1: The plots display the eigenvalues and numerical ranges of the elliptic Ginibre matrix ((A)--(D)) and the chiral elliptic Ginibre matrix ((E)--(H)). The red dots represent the eigenvalues together with the boundary of the droplet defined in \ref{['def of S for eGinUE']} and \ref{['def of S for Dirac']}, respectively. The solid black curves indicate the theoretical numerical ranges given in Theorem \ref{['Thm_elliptic and chiral']}. The blue dotted curves show numerically computed numerical ranges, which are in good agreement with the theoretical results. Here, $N=500$ for (A)--(D), while $N=250$ for (E)--(H).
• Figure 2: The same figure as in Figure \ref{['Fig_elliptic and chiral']} for the non-Hermitian Wishart ensemble. The solid black curve indicates the theoretical numerical range given in Theorem \ref{['Thm_NWishart']}. Here, $N=500$.
• Figure 3: The plots display the eigenvalues and simulated numerical ranges for products and powers of Ginibre matrices $Y_k$, where the matrices are normalised so that the associated droplet is the unit disc. In (A), we compare the product of two independent Ginibre matrices with the square of a single Ginibre matrix; in both cases, the numerical radius coincides with the value in \ref{['def of numerical radius for 2 products']}. Figures (B)--(D) show numerical ranges for various combinations of products and powers involving three Ginibre matrices. In all cases, the limiting numerical radius appears to be identical. Here, $N=500$.
• Figure 4: The simulated numerical range of the non-Hermitian Wishart matrix (blue solid curve), compared with the theoretical result in Theorem \ref{['Thm_NWishart']} (the region bounded by the green envelope), and with the elliptical ansatz \ref{['def of tilde E ellipse assumption']}. Here, $\alpha=2, \tau=0.8$ and $N=3000$.

Theorems & Definitions (19)

• Theorem 1.1: Numerical range of elliptic and chiral elliptic Ginibre matrices
• Remark 1: Outer and inner numerical radii
• Remark 2: Extension beyond the Gaussian setting
• Theorem 1.2: Numerical range of non-Hermitian Wishart matrix
• Remark 3: Maximally non-Hermitian case; products of two rectangular Ginibre matrices
• Remark 4: Hermitian limits
• Remark 5: Geometry of numerical range of non-Hermitian Wishart matrix
• Remark 6: Numerical ranges of products and powers of complex Ginibre matrices
• Proposition 2.1: Elliptic numerical range
• Lemma 2.2