Tensor

\mathbf{T} describes the stress experienced by a material at a given point. For any unit vector \mathbf{v} , the product \mathbf{T} \cdot \mathbf{v} is a vector, denoted \mathbf{T}(\mathbf{v}) , that quantifies the force per area along the plane perpendicular to \mathbf{v} . This image shows, for cube faces perpendicular to \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 , the corresponding stress vectors \mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3) along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.…

\mathbf{T} describes the stress experienced by a material at a given point. For any unit vector \mathbf{v} , the product \mathbf{T} \cdot \mathbf{v} is a vector, denoted \mathbf{T}(\mathbf{v}) , that quantifies the force per area along the plane perpendicular to \mathbf{v} . This image shows, for cube faces perpendicular to \mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 , the corresponding stress vectors \mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3) along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor. In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics, because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, etc.), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In applications, it is common to study situations in which a different tensor can occur at each point of an object. For example, the stress within an object may vary from one location to another. A family of tensors, that vary across space in this way, is a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor. \mathbf{\hat{e}}_i = \sum_{j=1}^n \mathbf{e}_j R^j_i = \mathbf{e}_j R^j_i . Here R j i are the entries of the change of basis matrix, and in the rightmost expression the summation sign was suppressed: this is the Einstein summation convention, which will be used throughout this article. The components v i of a column vector v transform with the inverse of the matrix R, \hat{v}^i = \left(R^{-1}\right)^i_j v^j, where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, w i , of a covector (or row vector), w, transform with the matrix R itself, \hat{w}_i = w_j R^j_i . This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript). As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array T that transforms under a change of basis matrix R = \left(R^j_i\right) by \hat{T} = R^{-1}TR . For the individual matrix entries, this transformation law has the form \hat{T}^{i'}_{j'} = \left(R^{-1}\right)^{i'}_i T^i_j R^j_{j'} so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1). Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above: \mathbf{v} = \hat{v}^i \,\mathbf{\hat{e}}_i = \left( \left(R^{-1}\right)^i_j {v}^j \right) \left( \mathbf _k R^k_i \right) = \left( \left(R^{-1}\right)^i_j R^k_i \right) {v}^j \mathbf _k = \delta_j^k {v}^j \mathbf _k = {v}^k \,\mathbf _k = {v}^i \,\mathbf _i , where \delta^k_j is the Kronecker delta, which functions similarly to the identity matrix, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will (commutativity), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like {v}^i \,\mathbf _i can immediately be seen to be geometrically identical in all coordinate systems. Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components (Tv)^i are given by (Tv)^i = T^i_j v^j . These components transform contravariantly, since \left(\widehat{Tv}\right)^{i'} = \hat{T}^{i'}_{j'} \hat{v}^{j'} = \left[ \left(R^{-1}\right)^{i'}_i T^i_j R^j_{j'} \right] \left[ \left(R^{-1}\right)^{j'}_k v^k \right] = \left(R^{-1}\right)^{i'}_i (Tv)^i . The transformation law for an order tensor with p contravariant indices and q covariant indices is thus given as, \hat{T}^{i'_1, \ldots, i'_p}_{j'_1, \ldots, j'_q} = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q} R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q}. Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type . The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type is also called a -tensor for short. This discussion motivates the following formal definition: {{blockquote|Definition. A tensor of type (p, q) is an assignment of a multidimensional array T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}] to each basis of an n-dimensional vector space such that, if we apply the change of basis \mathbf{f}\mapsto \mathbf{f}\cdot R = \left( \mathbf{e}_i R^i_1, \dots, \mathbf{e}_i R^i_n \right) then the multidimensional array obeys the transformation law T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}] R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} . }} The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. An equivalent definition of a tensor uses the representations of the general linear group. There is an action of the general linear group on the set of all ordered bases of an n-dimensional vector space. If \mathbf f = (\mathbf f_1, \dots, \mathbf f_n) is an ordered basis, and R = \left(R^i_j\right) is an invertible n\times n matrix, then the action is given by \mathbf fR = \left(\mathbf f_i R^i_1, \dots, \mathbf f_i R^i_n\right). Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let \rho be a representation of GL(n) on W (that is, a group homomorphism \rho: \text{GL}(n) \to \text{GL}(W) ). Then a tensor of type \rho is an equivariant map T: F \to W . Equivariance here means that T(FR) = \rho\left(R^{-1}\right)T(F). When \rho is a tensor representation of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds, and readily generalizes to other groups. T = T^{i_1\dots i_p}_{j_1\dots j_q}\; \mathbf{e}_{i_1}\otimes\cdots\otimes \mathbf{e}_{i_p}\otimes \boldsymbol{\varepsilon}^{j_1}\otimes\cdots\otimes \boldsymbol{\varepsilon}^{j_q}. Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type tensor. Moreover, the universal property of the tensor product gives a one-to-one correspondence between tensors defined in this way and tensors defined as multilinear maps. This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual: U \otimes V \cong\left(U^{* *}\right) \otimes\left(V^{* *}\right) \cong\left(U^{*} \otimes V^{*}\right)^{*} \cong \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right) The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right) and \operatorname{Hom}\left(U^{*} \otimes V^{*} ; \mathbb{F}\right) . Tensor products can be defined in great generality – for example, involving arbitrary modules over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space and its dual, as above. \hat{T}^{i'_1\dots i'_p}_{j'_1\dots j'_q}\left(\bar{x}^1, \ldots, \bar{x}^n\right) = \frac{\partial \bar{x}^{i'_1}}{\partial x^{i_1}} \cdots \frac{\partial \bar{x}^{i'_p}}{\partial x^{i_p}} \frac{\partial x^{j_1}}{\partial \bar{x}^{j'_1}} \cdots \frac{\partial x^{j_q}}{\partial \bar{x}^{j'_q}} T^{i_1\dots i_p}_{j_1\dots j_q}\left(x^1, \ldots, x^n\right). (S \otimes T)(v_1, \ldots, v_n, v_{n+1}, \ldots, v_{n+m}) = S(v_1, \ldots, v_n)T(v_{n+1}, \ldots, v_{n+m}), which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e., (S \otimes T)^{i_1\ldots i_l i_{l+1}\ldots i_{l+n}}_{j_1\ldots j_k j_{k+1}\ldots j_{k+m}} = S^{i_1\ldots i_l}_{j_1\ldots j_k} T^{i_{l+1}\ldots i_{l+n}}_{j_{k+1}\ldots j_{k+m}}. If is of type and is of type , then the tensor product has type . T = v_1\otimes w_1\otimes \alpha_1 + v_2\otimes w_2\otimes \alpha_2 +\cdots + v_N\otimes w_N\otimes \alpha_N. The contraction of T on the first and last slots is then the vector \alpha_1(v_1)w_1 + \alpha_2(v_2)w_2 + \cdots + \alpha_N(v_N)w_N. In a vector space with an inner product (also known as a metric) g, the term contraction is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a -tensor T^{ij} can be contracted to a scalar through T^{ij} g_{ij} (yet again assuming the summation convention). ===Computer vision and optics=== The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix. The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities: \frac{P_i}{\varepsilon_0} = \sum_j \chi^{(1)}_{ij} E_j + \sum_{jk} \chi_{ijk}^{(2)} E_j E_k + \sum_{jk\ell} \chi_{ijk\ell}^{(3)} E_j E_k E_\ell + \cdots. \! Here \chi^{(1)} is the linear susceptibility, \chi^{(2)} gives the Pockels effect and second harmonic generation, and \chi^{(3)} gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter. ===Machine learning=== The properties of tensors, especially tensor decomposition, have enabled their use in machine learning to embed higher dimensional data in artificial neural networks. This notion of tensor differs significantly from that in other areas of mathematics and physics, in the sense that a tensor is the same thing as a multidimensional array. Abstractly, a tensor belongs to tensor product of spaces, each of which has a fixed basis, and the dimensions of the factor spaces can be different. Thus, an example of a tensor in this context is a rectangular matrix. Just as a rectangular matrix has two axes, a horizontal and vertical axis to indicate the position of each entry, a more general tensor has as many axes as there are factors in the tensor product to which it belongs, and an entry of the tensor is referred to be a tuple of integers. The various axes have different dimensions in general. m = \int_\Omega \rho\, dx\,dy\,dz , where the Cartesian coordinates , , are measured in . If the units of length are changed into , then the numerical values of the coordinate functions must be rescaled by a factor of 100: x' = 100 x,\quad y' = 100y,\quad z' = 100 z . The numerical value of the density must then also transform by to compensate, so that the numerical value of the mass in kg is still given by integral of \rho\, dx\,dy\,dz . Thus \rho' = 100^{-3}\rho (in units of ). More generally, if the Cartesian coordinates , , undergo a linear transformation, then the numerical value of the density must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density, is a function of the variables , , (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold. A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition: T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] = \left|\det R\right|^{-w}\left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p} T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}] R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} . Here is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor. An example of a tensor density is the current density of electromagnetism. Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the rational representations of the general linear group. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still semisimple representations. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation, consisting of an with the transformation law (x, y) \mapsto (x + y\log \left|\det R\right|, y).

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