From Matrix Exponentials to Nonlinear Functions: A Common Structural View of Fourier Analysis, Hyperbolic Transformations, and Neural Learning (v1.1-en)

This research note explores how elementary linear generators, matrix exponentials, and projective readouts lead to familiar nonlinear functions. Beginning with Fourier synthesis and a tanh function unit, it derives matrix flows generated by a two-dimensional family satisfying A2 = εI for ε = −1, 0, or +1. The planar flows give circular, linear, and hyperbolic trajectories, while the projective coordinate Y/X obeys a Riccati equation and yields tan(t), t, and tanh(t). The note relates these representations to the logistic function and, as a separate linear interpretation of the same hyperbolic generator, a Lorentz boost. It then explains how a generator acts on tensor-product slots through the Leibniz rule, distinguishes that induced representation from tensors of higher input derivatives, and compares Fourier and tanh differentiation. A short neural-learning example and an appendix on spectral representations illustrate how the same structures can be viewed in different mathematical settings. The purpose is an accessible, checkable synthesis of established results and their relationships, with particular attention to the distinctions among dimension, tensor order, representation, and dynamics. 本研究札記從傅立葉模態與 tanh 函數單元出發,回到自然指數、線性微分方程與矩陣生成元,說明不同的函數表示如何由簡單的作用關係形成。藉由滿足 A2 = εI(ε = −1、0、+1)的二維生成元,逐步推導圓周、直線與雙曲線性流;再以分量比值 Y/X 作為射影座標,得到 Riccati 方程及 tan(t)、t、tanh(t) 等非線性函數。同時比較 Logistic 表示與 Lorentz boost 所展現的不同數學讀法。 後續以 Leibniz 乘積微分法則推導張量積上的誘導生成元,辨明其與輸入高階導數張量的區別,並比較傅立葉與 tanh 的導數結構。神經學習與頻譜表示僅作為延伸觀察窗口。全文強調基礎數學的逐步推導與跨領域關係的清晰闡釋,以協助讀者理解維度、張量階數、表示方式與演化作用之間的差異。 Files: the English PDF and a ZIP containing the LaTeX source, the Markdown manuscript (with one notation fix in Section 4.1, noted in the README), PNG/SVG figures, revision notes, and build settings. v1.1-en is the Zenodo release of manuscript version v0.2-r2 (2026-11-03). This note has not been peer reviewed.