The theory of linear dynamical systems is rich and thoroughly developed, yet most systems encountered in practice are nonlinear. This work aims to extend two classical tools from linear dynamics, diagonalization and spectral analysis, to the nonlinear setting. We begin by introducing a notion of diagonalization for nonlinear maps and subsequently study the associated Koopman operator acting on \(L^2\)
spaces, with \(\mu\) an invariant measure. Within this framework, we establish relationships between the spectra of diagonal nonlinear maps and those of diagonalizable nonlinear systems, allowing an n-dimensional problem to be decomposed into n one-dimensional problems. We then present a fully developed example of a nonlinear map in $C^1(\mathbb{R}^2)$ for which the diagonalization is completely characterized.