Multilinear algebra

Multilinear algebra is the study of functions and structures that are linear in multiple variables simultaneously. It extends the familiar framework of linear transformations to handle maps that take several vector inputs and combine them in ways that respect linearity in each argument independently.

The field encompasses tensors, which are the natural objects for encoding multilinear relationships, and bilinear forms, which pair vectors in a structured way. Core concepts include multilinear maps, tensor products, and exterior algebras—tools that reveal hidden symmetries in geometry and analysis.

Multilinear algebra permeates mathematics and physics. It underlies convolutional operations in machine learning, enables the formalism of differential forms in calculus on manifolds, and provides the language for describing physical properties like stress and strain. In modern transformer architectures, multilinear operations appear in attention mechanisms and higher-order parameter interactions.

The discipline bridges pure mathematics with applied domains: from Representation theory in algebra to Neural machine translation and large language models, where tensors encode rich, layered patterns in data.

Related

Tensor, Vector space, Algebra, Exterior algebra, Representation theory, Attention Is All You Need