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Tensor contraction is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the trace is a special case. It thereby reduces the total order of a tensor by two. It thereby reduces the total order of a tensor by two.
A (0,0) tensor is a number in the field . A (1,0) tensor is a vector. A (0,1) tensor is a covector. A (0,2) tensor is a bilinear form. An example is the metric tensor . A (1,1) tensor is a linear map.
In mathematics, a tensor is a certain kind of geometrical entity and array concept. It generalizes the concepts of scalar, vector and linear operator, in a way that is independent of any chosen frame of reference. For example, doing rotations over axis does not affect at all the properties of tensors, if a transformation law is followed.
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 ...
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.
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.