Multilinear algebra is a branch of mathematics that expands upon the principles of linear algebra. It extends the foundational theory of vector spaces by introducing the concepts of p-vectors and multivectors using Grassmann algebras.
In a vector space of dimension n, the focus is primarily on using vectors. However, Hermann Grassmann and others emphasized the importance of considering the structures of pairs, triplets, and general multi-vectors, which offer a more comprehensive perspective. With multiple combinatorial possibilities, the space of multi-vectors expands to 2n dimensions. The abstract formulation of the determinant is one direct application of multilinear algebra. Additionally, it finds practical use in studying the mechanical response of materials to stress and strain, involving various moduli of elasticity. The term "tensor" emerged to describe elements within the multi-linear space due to its added structure. This additional structure has made multilinear algebra significant in various fields of higher mathematics. However, despite Grassmann's early work in 1844 with his Ausdehnungslehre, which was also republished in 1862, it took time for the subject to gain acceptance, as ordinary linear algebra posed enough challenges on its own.
The concepts of multilinear algebra find applications in certain studies of multivariate calculus and manifolds, particularly in relation to the Jacobian matrix. Infinitesimal differentials encountered in single-variable calculus are transformed into differential forms in multivariate calculus, and their manipulation is carried out using exterior algebra.
Following Grassmann, developments in multilinear algebra were made by Victor Schlegel in 1872 with the publication of the first part of his System der Raumlehre and by Elwin Bruno Christoffel. Notably, significant advancements came through the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita, particularly in the form of absolute differential calculus within multilinear algebra. Marcel Grossmann and Michele Besso introduced this form to Albert Einstein, and in 1915, Einstein's publication on general relativity, explaining the precession of Mercury's perihelion, established multilinear algebra and tensors as important mathematical tools in physics.
In 1958, Nicolas Bourbaki included a chapter on multilinear algebra titled "Algèbra Multilinéair" in his series Éléments de mathématique, specifically within the book on algebra. The chapter covers topics such as bilinear functions, the tensor product of two modules, and the properties of tensor products.
The field of multilinear algebra has experienced less evolution in its subject matter compared to changes in its presentation over the years. The following pages provide additional information that is central to the topic:
There is also a glossary available for tensor theory.
Multilinear algebra concepts find applications in various areas, including: