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If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. [2] For instance, on space forms (i.e. spaces of constant sectional curvature) and two-dimensional smooth Riemannian manifolds, the Riemann curvature tensor has a simple expression in terms of the Kulkarni–Nomizu product of the metric = with itself; namely, if we denote by
In mathematics, 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.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
Topological tensor product. The tensor product, outer product and Kronecker product all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition. The outer ...
where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, α ∧ β = ε ∘ (α ⊗ β) ∘ Δ, where is the counit, as defined presently).
Geometrically, and are parallel if their geometric product is equal to their inner product, whereas and are perpendicular if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of ...
The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n -space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in ...