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In mathematics, multilinear algebra extends the methods of linear algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with 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 Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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.

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Historical background of the approach to multilinear algebra

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". History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 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 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. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition

One reason at the time was a new area of application, homological algebra. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector 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). In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, especially in Homological algebra and Algebraic topology, a Künneth theorem is a statement relating the homology of two objects to the The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined. In higher Mathematics, the Tor functors of Homological algebra are the Derived functors of the Tensor product functor

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 Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

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). Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group They instead applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism )

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. In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 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. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

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). The 1930s were described as an abrupt shift to more radical and conservative lifestyles as countries were struggling to find a solution to the Great Depression. Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. In a way this took the theory full circle, connecting once more the content of old and new viewpoints.

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. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, a bilinear map is a function of two arguments that is linear in each In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. 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 n × n Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after In Mathematics, the modern Component-free approach to the theory of Tensors views tensors initially as Abstract objects expressing some definite type of In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional 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 In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematics, especially in the area of Abstract algebra known as Ring theory, a free algebra is the noncommutative analogue of a Polynomial ring In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational In Mathematics, a symmetric tensor is a Tensor that is invariant under a Permutation of its vector arguments In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space This is a glossary of tensor theory. For expositions of tensor theory from different points of view see Tensor Classical treatment of tensors

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:

Contravariant and covariant tensors A contravariant tensor of order 1(T^i is defined as \bar{T}^i = T^r\frac{\partial \bar{x}^i}{\partial x^r} A dyadic tensor in Multilinear algebra is a second rank Tensor written in a special notation formed by juxtaposing pairs of vectors i Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped In Mathematics, Clifford algebras are a type of Associative algebra. In Physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as Improper rotations In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the In Linear algebra, the outer product typically refers to the tensor product of two vectors. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic
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