the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product...
23 KB (4,160 words) - 15:09, 12 January 2024
mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear...
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manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on tensor algebras. Here, there is much...
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mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map...
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two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense...
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mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring...
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algebra concepts find applications in various areas, including: Classical treatment of tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket...
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multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor index notation. Order of a tensor The components of a tensor with respect...
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category of R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more...
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In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold...
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enveloping algebras. The construction proceeds by first building the tensor algebra of the underlying vector space of the Lie algebra. The tensor algebra is simply...
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In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...
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of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication...
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diagonalization. The free algebra generated by V may be written as the tensor algebra ⨁n≥0 V ⊗ ⋯ ⊗ V, that is, the direct sum of the tensor product of n copies...
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symmetric algebra. As the symmetric algebra of a vector space is a quotient of the tensor algebra, an element of the symmetric algebra is not a tensor, and...
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and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used...
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In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v...
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tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor...
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tensor algebra T ( g ) {\displaystyle T({\mathfrak {g}})} from it. The tensor algebra is a free algebra: it simply contains all possible tensor products...
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Penrose graphical notation (redirect from Tensor diagram notation)
essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting...
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In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting...
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multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal...
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Multilinear algebra Tensor Classical treatment of tensors Component-free treatment of tensors Gamas's Theorem Outer product Tensor algebra Exterior algebra Symmetric...
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Outer product (section The outer product of tensors)
referred to as their tensor product, and can be used to define the tensor algebra. The outer product contrasts with: The dot product (a special case of...
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at p. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence...
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Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products...
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theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general...
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Universal property (section Tensor algebras)
property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier...
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(indicial) tensor manipulation, ctensor for component-defined tensors, and atensor for algebraic tensor manipulation. Tensor is an R package for basic tensor operations...
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metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g(v, v) > 0 for...
56 KB (8,866 words) - 08:52, 9 August 2024