Eigenvalue collisions for periodic matrix families associated with Ginibre matrices

Abstract

We study, count and locate the exceptional points where eigenvalues collide for certain families of matrices $$R(s,t) = \cos(s \pi / 2)C + \sin(s \pi / 2)U(t), \quad s,t \in [0,1]$$ where $C$ is a realization of a Ginibre random matrix, or a closely related matrix, and $U(t)$ is a $t$-periodic diagonal matrix whose eigenvalues move along the unit circle or other parameterized curves (simple or not). To do this, we track eigenvalues continuously along simple curves, searching for eigenvalue discrepancies after looping, indicating such collision-events inside said curves. Although stochastic in nature, the collision count for the circle cases repeatedly yielded $N(N-1)$ under fairly general circumstances, where $N$ is the matrix dimension. More collisions were observed for the elliptic generalization and for other curves. We include a package to compute and visualize these results.