Abstract
This article provides an advanced overview of the structural, analytical, and geometric bridges connecting multidimensional linear algebra and multilinear harmonic analysis. We explore the foundational duality between metric tensor products and multilinear forms, examine modern multilinear Calderón–Zygmund theory, review phase-space wave packet decompositions underlying the bilinear Hilbert transform, detail the role of multilinear Kakeya and Brascamp–Lieb inequalities in ℓ2(Lp) decoupling theory, and synthesize their applications to nonlinear partial differential equations through fractional Leibniz rules and multi-parameter analysis.