Multilinear algebra is the study offunctions with multiplevector-valuedarguments, with the functions beinglinear maps with respect to each argument. It involves concepts such asmatrices,tensors,multivectors,systems of linear equations,higher-dimensional spaces,determinants,inner andouter products, anddual spaces. It is a mathematical tool used inengineering,machine learning,physics, andmathematics.[1]
While many theoretical concepts and applications involvesingle vectors, mathematicians such asHermann Grassmann considered structures involving pairs, triplets, andmultivectors that generalizevectors. With multiple combinational possibilities, the space ofmultivectors expands to 2n dimensions, wheren is the dimension of the relevant vector space.[2] Thedeterminant can be formulated abstractly using the structures of multilinear algebra.
Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli ofelasticity. The term "tensor" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with hisAusdehnungslehre, which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time.
The concepts of multilinear algebra find applications in certain studies ofmultivariate calculus andmanifolds, particularly concerning theJacobian matrix.Infinitesimal differentials encountered in single-variable calculus are transformed intodifferential forms inmultivariate calculus, and their manipulation is carried out usingexterior algebra.[3]
Following Grassmann, developments in multilinear algebra were made byVictor Schlegel in 1872 with the publication of the first part of hisSystem der Raumlehre[4] and byElwin Bruno Christoffel. Notably, significant advancements came through the work ofGregorio Ricci-Curbastro andTullio Levi-Civita,[5] particularly in the form ofabsolute differential calculus within multilinear algebra.Marcel Grossmann andMichele Besso introduced this form toAlbert Einstein, and in 1915, Einstein's publication ongeneral relativity, explaining theprecession 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 "Algèbre Multilinéaire" in their seriesÉléments de mathématique, specifically within the algebra book. The chapter covers topics such as bilinear functions, thetensor product of twomodules, and the properties of tensor products.[6]
Multilinear algebra concepts find applications in various areas, including:
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