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Diagonalization of nonlinear functions in finite-dimensional spaces and the case of ℝ2

  • Roger Arnau
  • Juan Carlos Cortés López
  • Álvaro González Cortés
  • Enrique Alfonso Sánchez Pérez
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Abstract

This paper introduces a new theoretical framework for the diagonalization of nonlinear functions defined in finite-dimensional real Euclidean spaces. Extending classical results from linear algebra, we provide a unified setting to determine when a nonlinear map can be represented in diagonal form via a change of basis. Due to the complexity of the equations involved, the final part of the paper focuses primarily on the two-dimensional case, for which clear characterizations can be obtained. We also illustrate how these theoretical findings can be applied to classical contexts, such as differential equations, dynamical systems, and the explicit computation of higher-order compositions and inverses of functions.

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Cite

BibTeXRIS
BibTeX
@misc{arnau2026diagonalization,
  title = {{Diagonalization of nonlinear functions in finite-dimensional spaces and the case of $\mathbb R^2$}},
  author = {Roger Arnau and Juan Carlos Cortés López and Álvaro González Cortés and Enrique Alfonso Sánchez Pérez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.16437},
  eprint = {2606.16437},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.16437},
}
RIS
TY  - UNPB
TI  - Diagonalization of nonlinear functions in finite-dimensional spaces and the case of ℝ^2
AU  - Roger Arnau
AU  - Juan Carlos Cortés López
AU  - Álvaro González Cortés
AU  - Enrique Alfonso Sánchez Pérez
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2606.16437
UR  - https://arxiv.org/abs/2606.16437
AB  - This paper introduces a new theoretical framework for the diagonalization of nonlinear functions defined in finite-dimensional real Euclidean spaces. Extending classical results from linear algebra, we provide a unified setting to determine when a nonlinear map can be represented in diagonal form via a change of basis. Due to the complexity of the equations involved, the final part of the paper focuses primarily on the two-dimensional case, for which clear characterizations can be obtained. We also illustrate how these theoretical findings can be applied to classical contexts, such as differential equations, dynamical systems, and the explicit computation of higher-order compositions and inverses of functions.
ER  -