To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Multilinear algebra

From Wikipedia, the free encyclopedia

Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebras.

Origin

In a vector space of dimension n, normally only the vectors are used. However; according to Hermann Grassmann and others, this presumption misses the complexity of considering the structures of pairs, triples, and general multivectors. With several combinatorial possibilities, the space of multivectors has 2n dimensions. The abstract formulation of the determinant is the most immediate application. Multilinear algebra also has applications in the mechanical study of material response to stress and strain with various moduli of elasticity. This practical reference led to the use of the word tensor to describe the elements of the multilinear space. The extra structure in a multilinear space has led it to play an important role in various studies in higher mathematics. Though Grassmann started the subject in 1844 with his Ausdehnungslehre, which was also republished in 1862, his work was slow to find acceptance as ordinary linear algebra provided sufficient challenges to comprehension.

The topic of multilinear algebra is applied in some studies of multivariate calculus and manifolds where the Jacobian matrix comes into play. The infinitesimal differentials of single variable calculus become differential forms in multivariate calculus, and their manipulation is done with exterior algebra.

After Grassmann, developments in multilinear algebra were made in 1872 by Victor Schlegel when he published the first part of his System der Raumlehre, and by Elwin Bruno Christoffel. A major advance in multilinear algebra came in the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita (see references). It was the absolute differential calculus form of multilinear algebra that Marcel Grossmann and Michele Besso introduced to Albert Einstein. The publication in 1915 by Einstein of a general relativity explanation for the precession of the perihelion of Mercury, established multilinear algebra and tensors as physically important mathematics.

Use in algebraic topology

Around the middle of the 20th century the study of tensors was reformulated more abstractly. The Bourbaki group's treatise Multilinear Algebra was especially influential—in fact, the term multilinear algebra may have originated there.[citation needed]

One reason at the time was a new area of application, homological algebra. The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. The computation of the homology groups of the product of two topological spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined.

The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to de Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product.

The resulting rather severe write-up of the topic, by Bourbaki, entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups), and instead, applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. Since this leads to a much cleaner treatment, there was probably no going back in purely mathematical terms. (Strictly, the universal property approach was invoked; this is somewhat more general than category theory, and the relationship between the two as alternate ways was also being clarified, at the same time.)

Indeed, what was done is almost precisely to explain that tensor spaces are the constructions required to reduce multilinear problems to linear problems. This purely algebraic attack conveys no geometric intuition.

By re-expressing problems in terms of multilinear algebra, there is a clear and well-defined "best solution": the constraints the solution exerts are exactly those needed in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to co-ordinate systems. In the category-theoretic jargon, everything is entirely natural.

Topics in multilinear algebra

The subject matter of multilinear algebra has evolved less than the presentation down the years. Here are further pages centrally relevant to it:

There is also a glossary of tensor theory.

Applications

Some of the ways in which multilinear algebra concepts are applied:

References

  • Hermann Grassmann (2000) Extension Theory, American Mathematical Society. Translation by Lloyd Kannenberg of the 1862 Ausdehnungslehre.
  • Fleming, Wendell H. (1977). "Exterior algebra and differential calculus". Functions of several variables (2nd ed.). New York: Springer-Verlag. ISBN 0-387-90206-6. OCLC 2401829.
  • 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
  • Ronald Shaw (1983) "Multilinear algebra and group representations", volume 2 of Linear Algebra and Group Representations, Academic Press ISBN 0-12-639202-1.
This page was last edited on 20 May 2022, at 13:17
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.