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This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2.
C++ 2008 3.4.0 / 08.2021 Free MPL2: Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms. Fastor [5] R. Poya, A. J. Gil and R. Ortigosa C++ 2016 0.6.4 / 06.2023 Free MIT License: Fastor is a high performance tensor (fixed multi-dimensional array) library for modern C++. GNU Scientific ...
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 ...
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.
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
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.
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):
be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors of W in the product). Then the pullback Φ ∗ F of F by Φ is a multilinear form on V defined by precomposing F with Φ. More precisely, given vectors v 1, v 2, ..., v s in V, Φ ∗ F is defined by ...