The eigenvalues of a matrix are often of interest in math and science, particularly for describing the solutions to linear differential equations. In studying systems with delay, damping or radiation, however, the nonlinear eigenproblem arises: for an analytic matrix-valued function $T : \Omega \rightarrow \mathbb{C}^{n \times n}$, the nonlinear eigenproblem is to find $\lambda \in \Omega$ such that the matrix $T(\lambda)$ is singular.

Tools such as Gershgorin’s theorem are useful for localizing eigenvalues of a matrix, but such results have only been extended to special cases of the nonlinear eigenproblem (where $T$ is polynomial, for instance). With examples from delay differential equations and quantum scattering, I will illustrate several new localization results for the general nonlinear eigenproblem.