Reconstructing the Link Between Linear Algebra and Trigonometry: Matrix-Free Spectral Theory from Phase Algebra

Abstract The historical development of modern mathematics has produced a conceptual separation between linear algebra and trigonometry. Linear algebra evolved toward abstract spectral theory and operator methods, while trigonometric functions were relegated to the domain of special functions in analysis, obscuring their intrinsic spectral origin and their connection to phase structure and linear transformations. In this work, we present a unified algebraic framework that reconnects linear algebra and trigonometry at a fundamental level. Starting from a phase algebra generated by an element satisfying a cubic relation, we show that spectral decomposition, eigenvalue structure, and idempotent projections arise intrinsically, without reference to matrices, determinants, or coordinate representations. A distinguished phase combination satisfies a quadratic minimal polynomial, yielding a two-channel spectral structure governed by fixed eigenvalues. Within this framework, trigonometric functions emerge naturally as functional realizations of spectral decomposition through the exponential map associated with the phase algebra, rather than as independently postulated analytic objects. This construction demonstrates that trigonometry is fundamentally a spectral phenomenon, with the classical complex formulation appearing as a special two-phase case. The resulting theory precedes the introduction of norms, inner products, or probabilistic interpretations, applies beyond the quantum-mechanical context, provides a matrix-free realization of spectral theory, and establishes a direct bridge between phase algebra, trigonometric structure, and abstract linear analysis.