Multilinear algebra

Multilinear algebra is the study of functions with multiple vector-valued arguments, which are linear maps with respect to each argument. Concepts such as matrices, vectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces emerge naturally in the mathematics of multilinear functions. Multilinear algebra is a foundational mathematical tool in engineering, machine learning, physics, and mathematics.

Origin

While many theoretical concepts and applications are concerned with single vectors, mathematicians such as Hermann Grassmann considered the structures involving pairs, triplets, and general multi-vectors that generalize vectors. With multiple combinational possibilities, the space of multi-vectors expands to 2n dimensions, where n is the dimension of the relevant vector space.[1] The determinant 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 of elasticity. The term "tensor" describes elements within the multilinear space due to its added structure. Despite Grassmann's early work in 1844 with his Ausdehnungslehre, 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 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.[2]

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[3] and by Elwin Bruno Christoffel. Notably, significant advancements came through the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita,[4] 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 "Algèbre Multilinéaire" 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.[5]

Topics in multilinear algebra

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.

Applications

Multilinear algebra concepts find applications in various areas, including:

References
  1. Grassmann, Hermann (2000) [1862]. Extension Theory [Die Ausdehnungslehre]. Translated by Kannenberg, Lloyd. American Mathematical Society. ISBN 978-0-8218-9049-3.
  2. 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.
  3. 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.
  4. 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.
  5. Nicolas Bourbaki (1958) Algèbra Multilinéair, chapter 3 of book 2 Algebra, in Éléments de mathématique, Paris: Hermann
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