The eigenvector-dependent nonlinear eigenvalue problem (NEPv), also known as nonlinear eigenvector problem, is a special type of eigenvalue problem where we seek to find \(k\) eigenpairs of a Hermitian matrix function \(H : \mathbb{C}^{n,k} \to \mathbb{C}^{n,n}\) that depends nonlinearly on the eigenvectors itself. That is, we have to find \(V \in \mathbb{C}^{n,k}\) with orthonormal columns and \(\Lambda \in \mathbb{C}^{k,k}\) such that \(H(V) V = V \Lambda\). NEPv arise in a variety of applications, most notably in Kohn-Sham density theory and data science applications such as robust linear discriminant analysis. This talk is concerned with solving NEPv by viewing it as a set of nonlinear matrix equations and using an inexact Newton method on a matrix level. In this setting, Newton's method is applied using the Fréchet derivative and exploiting the structure of the problem by using a global GMRES-approach to solve the Newton-update equation efficiently.