Multilinear algebra

Multilinear algebrais the study offunctionswith multiplevector-valuedarguments,with the functions beinglinear mapswith respect to each argument. It involves concepts such asmatrices,tensors,multivectors,systems of linear equations,higher-dimensional spaces,determinants,innerandouterproducts, anddual spaces.It is a mathematical tool used inengineering,machine learning,physics,andmathematics.^[1]

Origin[edit]

While many theoretical concepts and applications involvesingle vectors,mathematicians such asHermann Grassmannconsidered structures involving pairs, triplets, andmultivectorsthat generalizevectors.With multiple combinational possibilities, the space ofmultivectorsexpands to 2ⁿdimensions, wherenis the dimension of the relevant vector space.^[2]Thedeterminant can be formulated abstractlyusing 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 calculusandmanifolds,particularly concerning theJacobian matrix.Infinitesimal differentialsencountered in single-variable calculus are transformed intodifferential formsinmultivariate calculus,and their manipulation is carried out usingexterior algebra.^[3]

Following Grassmann, developments in multilinear algebra were made byVictor Schlegelin 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-CurbastroandTullio Levi-Civita,^[5]particularly in the form ofabsolute differential calculuswithin multilinear algebra.Marcel GrossmannandMichele Bessointroduced 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 Bourbakiincluded a chapter on multilinear algebra titled "Algèbre Multilinéaire"in his seriesÉléments de mathématique,specifically within the algebra book. The chapter covers topics such as bilinear functions, thetensor productof twomodules,and the properties of tensor products.^[6]

Applications[edit]

Multilinear algebra concepts find applications in various areas, including:

References[edit]

^Pandey, Divyanshu; Venugopal, Adithya; Leib, Harry (2024)."Linear to multi-linear algebra and systems using tensors".Frontiers in Applied Mathematics and Statistics.9.arXiv:2304.10658.doi:10.3389/fams.2023.1259836.ISSN 2297-4687.
^Grassmann, Hermann (2000) [1862].Extension Theory[Die Ausdehnungslehre]. Translated by Kannenberg, Lloyd.American Mathematical Society.ISBN 978-0-8218-9049-3.
^Fleming, Wendell H. (1977)."Exterior algebra and differential calculus".Functions of Several Variables.Undergraduate Texts in Mathematics (2nd ed.). Springer. pp. 275–320.doi:10.1007/978-1-4684-9461-7_7.ISBN 978-1-4684-9461-7.OCLC 2401829.
^Schlegel, Victor (2018).System der Raumlehre: Nach den Prinzipien der Grassmann'schen Ausdehnungslehre und als Einleitung in Dieselbe; Geometrie; Die Gebiete des Punktes, der Geraden, der Ebene.Forgotten Books.ISBN 978-0-364-22177-8.
^Ricci-Curbastro, Gregorio;Levi-Civita, Tullio(1900)."Méthodes de calcul différentiel absolu et leurs applications".Mathematische Annalen.54(1): 125–201.doi:10.1007/BF01454201.ISSN 1432-1807.S2CID 120009332.
^Nicolas Bourbaki(1958)Algèbra Multilinéair,chapter 3 of book 2Algebra,inÉléments de mathématique,Paris: Hermann

Greub, W. H. (1967)Multilinear Algebra,Springer
Douglas Northcott(1984)Multilinear Algebra,Cambridge University PressISBN 0-521-26269-0
Shaw, Ronald (1983).Multilinear algebra and group representations.Vol. 2.Academic Press.ISBN 978-0-12-639202-9.OCLC 59106339.

[1] Pandey, Divyanshu; Venugopal, Adithya; Leib, Harry (2024)."Linear to multi-linear algebra and systems using tensors".Frontiers in Applied Mathematics and Statistics.9.arXiv:2304.10658.doi:10.3389/fams.2023.1259836.ISSN 2297-4687.

[2] Grassmann, Hermann (2000) [1862].Extension Theory[Die Ausdehnungslehre]. Translated by Kannenberg, Lloyd.American Mathematical Society.ISBN 978-0-8218-9049-3.

[3] Fleming, Wendell H. (1977)."Exterior algebra and differential calculus".Functions of Several Variables.Undergraduate Texts in Mathematics (2nd ed.). Springer. pp. 275–320.doi:10.1007/978-1-4684-9461-7_7.ISBN 978-1-4684-9461-7.OCLC 2401829.

[4] Schlegel, Victor (2018).System der Raumlehre: Nach den Prinzipien der Grassmann'schen Ausdehnungslehre und als Einleitung in Dieselbe; Geometrie; Die Gebiete des Punktes, der Geraden, der Ebene.Forgotten Books.ISBN 978-0-364-22177-8.

[5] Ricci-Curbastro, Gregorio;Levi-Civita, Tullio(1900)."Méthodes de calcul différentiel absolu et leurs applications".Mathematische Annalen.54(1): 125–201.doi:10.1007/BF01454201.ISSN 1432-1807.S2CID 120009332.

[6] Nicolas Bourbaki(1958)Algèbra Multilinéair,chapter 3 of book 2Algebra,inÉléments de mathématique,Paris: Hermann

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Origin[edit]

Applications[edit]

See also[edit]

References[edit]