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...

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...

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...

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...

manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on tensor algebras. Here, there is much...

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...

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...

algebra concepts find applications in various areas, including: Classical treatment of tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket...

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...

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...

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...

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...

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...

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...

multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor index notation. Order of a tensor The components of a tensor with respect...

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...

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...

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...

essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting...

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...

tensor algebra T ( g ) {\displaystyle T({\mathfrak {g}})} from it. The tensor algebra is a free algebra: it simply contains all possible tensor products...

In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting...

linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components...

Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products...

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...

Multilinear algebra Tensor Classical treatment of tensors Component-free treatment of tensors Gamas's Theorem Outer product Tensor algebra Exterior algebra Symmetric...

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...

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...

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...

category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra Free object Noncommutative...