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Encyclopedia > Multilinear algebra

In mathematics, multilinear algebra 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 concept of a tensor and develops the theory of 'tensor spaces'. In applications, numerous types of tensors arise. The theory tries to be comprehensive, with a corresponding range of spaces and an account of their relationships. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

## Historical background of the approach to multilinear algebra GA_googleFillSlot("encyclopedia_square");

The subject itself has various roots going back to the mathematics of the nineteenth century, in what was then called tensor analysis, or the "tensor calculus of tensor fields". It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics. 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 was probably coined there. For more technical Wiki articles on tensors, see the section later in this article. ... For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ...

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 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. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... A torus. ... In mathematics, the KÃ¼nneth theorem of algebraic topology describes the singular homology of the cartesian product X Ã— Y of two topological spaces, in terms of singular homology groups Hi(X, R) and Hj(X, R). ... The Tor functors are the derived functors of the tensor product functor in mathematics. ...

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. Hermann GÃ¼nther Grassmann (April 15, 1809, Stettin â€“ September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

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). They 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.) Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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.

Its benefit is that 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 you need 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.

## Conclusion on the abstract approach

In principle the abstract approach can recover everything done via the traditional approach. In practice this may not seem so simple. On the other hand the notion of natural is consistent with the general covariance principle of general relativity. The latter deals with tensor fields (tensors varying from point to point on a manifold), but covariance asserts that the language of tensors is essential to the proper formulation of general relativity. This article or section is in need of attention from an expert on the subject. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

Some decades later the rather abstract view coming from category theory was tied up with the approach that had been developed in the 1930s by Hermann Weyl (in his celebrated and difficult book The Classical Groups). In a way this took the theory full circle, connecting once more the content of old and new viewpoints. This article or section is missing references or citation of sources. ... Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ...

## Content of multilinear algebra

The content of multilinear algebra has changed much less than the presentation, down the years. Here are further pages centrally relevant to it:

There is also a glossary of tensor theory. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ... Note: This is a fairly abstract mathematical approach to tensors. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ... In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ... The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ... A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... This is a glossary of tensor theory. ...

## From the point of view of applications

Consult these articles for some of the ways in which multilinear algebra concepts are applied, in various guises:

Results from FactBites:

 Michael Tsatsomeros' CV (886 words) Linear and Multilinear Algebra, 50 : 151-165, 2002. Linear and Multilinear Algebra, 45 : 75-97, 1998. Linear and Multilinear Algebra, 41 : 199-210, 1996.
 Publications of Russell Merris (873 words) Grone, Linear & Multilinear Algebra 39 (1995), 133-136. Bapat, Linear & Multilinear Algebra 42 (1997), 159-167. Lorenzini, Linear & Multilinear Algebra 47 (2000), 281-306.
More results at FactBites »

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