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Encyclopedia results for Tensor product

Tensor product





Encyclopedia results for Tensor product

  1. Tensor product

    In mathematics , the tensor product , denoted by , may be applied in different contexts to vector space ... operation . In some contexts, this product is also referred to as outer product . The term tensor product is also used in relation to monoidal category monoidal categories . Tensor product of vector spaces The tensor product V     W of two vector space s V and W over a field mathematics field ... generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space math V otimes W F V times W R. math It is also called the tensor product space ... . The tensor product of two elements v and w is the equivalence class e sub v , w sub R of e sub ... form a basis for V     W . The dimension of the tensor product therefore is the product of dimensions ... of the tensor product space is not a pure tensor, but rather a finite linear combination of pure ... to express an element of a tensor product is called the tensor rank , not to be confused with tensor ... The tensor product also operates on linear map s between vector spaces. Specifically, given two linear maps S V X and T W Y between vector spaces, the tensor product of the two linear maps S and T is a linear ... w S v otimes T w . math In this way, the tensor product becomes a bifunctor from the category of vector ... product of two matrices is the matrix of the tensor product of the two corresponding linear maps under a standard choice of bases of the tensor products see Kronecker product Relation to the abstract tensor product the article on Kronecker products . More than two vector spaces The construction and the universal property of the tensor product can be extended to allow for more than two vector .... The tensor product V sub 1 sub     V sub 2 sub     V sub 3 sub is defined along ... map. The tensor product is uniquely characterized by this property, up to a unique isomorphism. This construction ..., the tensor product of an arbitrary indexed family V sub i sub , i     I , is defined to be universal ...   more details



  1. Tensor product of algebras

    ref improve date May 2010 In mathematics , the tensor product of two algebra ring theory R algebras is also an R algebra. This gives us a tensor product of algebras . The special case R Z gives us a tensor product of rings , since rings may be regarded as Z algebras. Let R be a commutative ring and let A and B be algebra ring theory R algebras . Since A and B may both be regarded as module mathematics R modules , we may form their tensor product of modules tensor product math A otimes R B math which is also an R module. We can give the tensor product the structure of an algebra by defining ref Kassel 1995 , Google books quote id S1KE pToY98C page 32 text we put an algebra structure on the tensor product p. 32 . ref math a 1 otimes b 1 a 2 otimes b 2 a 1a 2 otimes b 1b 2 math and then extending by linearity to all of nowrap A &thinsp sub R sub &thinsp B . This product is easily seen to be R bilinear, associative, and unital with an identity element given by nowrap 1 sub A sub &thinsp &thinsp 1 sub B sub , ref Kassel 1995 , Google books quote id S1KE pToY98C page 32 text Its unit is p. 32 . ref where 1 sub A sub and 1 sub B sub are the identities of A and B . If A and B are both commutative then the tensor product is as well. The tensor product turns the category mathematics category ... b math These maps make the tensor product a coproduct in the category of commutative R algebras. The tensor product is not the coproduct in the category of all R algebras. There the coproduct is given by a more general free product of algebras . The tensor product of algebras is of constant use ... itself, it is possible to construct the tensor product of rings tensor product of R over a ring S with another ... of scalars Tensor product of modules Tensor product of fields Notes references References Citation ... publisher Springer isbn 9780387943701 . DEFAULTSORT Tensor Product Of Algebras Category ... pullback category theory pullback s of affine scheme s, otherwise known as fiber product s. Stuf ...   more details



  1. Tensor product of graphs

    Image Graph tensor product.svg thumb 360px The tensor product of graphs. In graph theory , the tensor product G H of graphs G and H is a graph such that the vertex set of G H is the Cartesian product V ... is adjacent with v . The tensor product is also called the direct product , categorical product , cardinal product , relational product , Kronecker product , weak direct product , or conjunction . As an operation on binary relations, the tensor product was introduced by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica 1912 . It is also equivalent to the Kronecker product ... sometimes used to represent another construction known as the Cartesian product of graphs , but more commonly refers to the tensor product. The cross symbol shows visually the two edges resulting from the tensor product of two edges. ref citation title Graph symmetry algebraic methods and applications ...?id tIaXdII8egC&pg PA116 . ref Examples The tensor product G K sub 2 sub is a bipartite graph , called ... graph a complete bipartite graph K sub n , n sub minus a perfect matching . The tensor product ... row or column of the grid. Properties The tensor product is the Product category theory category theoretic product in the category of graphs and graph homomorphism s. That is, there is a homomorphism ... the homomorphisms to G and H . The adjacency matrix of G H is the Kronecker product tensor product of the adjacency matrices of G and H . If a graph can be represented as a tensor product, then there may be multiple different representations tensor products do not satisfy unique factorization ... time algorithm for recognizing tensor product graphs and finding a factorization of any such graph. If either G or H is bipartite , then so is their tensor product. G H is connected if and only if both .... The Hedetniemi conjecture gives a formula for the chromatic number of a tensor product. See also Graph product Strong product of graphs Notes reflist References cite journal author Imrich, W. title ...   more details



  1. Tensor product network

    Orphan date December 2009 A tensor product network , in neural network s, is a network that exploits the properties of tensor s to model associative concepts such as variable computer science variable assignment. Orthonormality Orthonormal vectors are chosen to model the ideas such as variable names and target assignments , and the tensor product of these Euclidean vector vector s construct a network whose mathematical properties allow the user to easily extract the association from it. See also Neural network Neuroscience Unreferenced date December 2009 DEFAULTSORT Tensor Product Network Category Neural networks Sci stub ...   more details



  1. Tensor product of quadratic forms

    Unreferenced date February 2008 The tensor product of quadratic form s is most easily understood when one views the quadratic forms as quadratic spaces . So, if V, q sub 1 sub and W, q sub 2 sub are quadratic spaces, which V,W vector spaces, then the tensor product is a quadratic form q on the Tensor product Tensor product of vector spaces tensor product of vector spaces math V otimes W math . It is defined in such a way that for math v otimes w in V otimes W math we have math q v otimes w q 1 v q 2 w math . In particular, if we have diagonalizations of our quadratic forms which is always possible when the characteristic algebra characteristic is not 2 such that math q 1 cong langle a 1, ... , a n rangle math math q 2 cong langle b 1, ... , b m rangle math then the tensor product has diagonalization math q 1 otimes q 2 q cong langle a 1b 1, a 1b 2, ... a 1b m, a 2b 1, ... , a 2b m , ... , a nb 1, ... a nb m rangle. math Category Quadratic forms Category Tensors algebra stub ...   more details



  1. Topological tensor product

    In mathematics , there are usually many different ways to construct a topological tensor product of two ... theory of tensor product s see Tensor product of Hilbert spaces , but for general Banach space s or locally convex topological vector space the theory is notoriously subtle. Tensor products of Hilbert spaces Main Tensor product of Hilbert spaces The algebraic tensor product of two Hilbert spaces A and B ... is a Hilbert space A B , called the Hilbert space tensor product of A and B . If the vectors a sub ... j sub form an orthonormal basis of A B . Cross norms and tensor products of Banach spaces We shall use the notation from harv Ryan 2002 in this section. The obvious way to define the tensor product ... tensor product, then take the completion in this norm. The problem is that there is more than one natural way to define a norm on the tensor product. If A and B are Banach spaces the algebraic tensor product of A and B means the tensor product of A and B as vector spaces and is denoted by math A otimes B math . The algebraic tensor product math A otimes B math consists of all finite sums math x ... a cross norm p on the algebraic tensor product math A otimes B math is a norm satisfying the conditions ... of the algebraic tensor product in these two norms are called the projective and injective tensor products, and are denoted by math A hat otimes pi B math and math A hat otimes varepsilon B math . The norm used for the Hilbert space tensor product is not equal to either of these norms in general. Some authors denote it by , so the Hilbert space tensor product in the section above would be math ... by &alpha on math A otimes B math and on the completed tensor product math A hat otimes alpha B ... on the algebraic tensor product A B , and by choosing one cross norm from each family we get some cross ... one sensible tensor product of A and B . This property characterizes nuclear spaces. See also Hilbert ... Tensor product of Hilbert spaces Fredholm kernel Projective topology References citation last Ryan ...   more details



  1. Tensor product of Hilbert spaces

    In mathematics , and in particular functional analysis , the tensor product of Hilbert space s is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert space is another Hilbert space. Roughly speaking, the tensor product is the complete metric space completion of the ordinary tensor product. This is a special case of a topological tensor product ..., and therefore a topology, on the tensor product that arise naturally from those of the factors. Let  ..., cdot rangle 1 math and math langle cdot, cdot rangle 2 math , respectively. Construct the tensor product ... product s. We can turn this vector space tensor product into an inner product space by defining math ... tensor product. Finally, take the complete space Completion completion under this inner product. The resulting Hilbert space is the tensor product of   H sub 1 sub and  H sub 2 sub . Explicit construction The tensor product can also be defined without appealing to the metric space completion. If H sub 1 sub and H sub 2 sub are two Hilbert spaces, one associates to every simple tensor product ... tensor product of H sub 1 sub and H sub 2 sub , that is isometrically and linearly isomorphic to HS H sub 1 sub sup &lowast sup , H sub 2 sub . Universal property The Hilbert tensor product math H H ... the tensor product H up to isomorphism. The same universal property, with obvious modifications, also applies for the tensor product of any finite number of Hilbert spaces. Infinite tensor ... of unit vectors in these Hilbert spaces then the incomplete tensor product or Guichardet tensor product ... math . Then the von Neumann tensor product of the von Neumann algebras is the strong completion of the set ... of the tensor product is the product as cardinal number s of the Hilbert dimensions. Examples ... in the construction of the Hilbert space tensor product. Similarly, we can show that L sup 2 sup ... of both particles is described by the tensor product of  H sub 1 sub and  H sub 2 sub . For example ...   more details



  1. Tensor product of modules

    In mathematics , the tensor product of modules is a construction that allows arguments about bilinear ... s . The module construction is analogous to the construction of the tensor product of vector space ... ring , with result an abelian group . Tensor products are important in areas of abstract algebra , homological algebra , algebraic topology and algebraic geometry . The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operator linear operations . The tensor product of an algebra and a module can be used for extension 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 ... Let M , N and R be as in the previous section. The tensor product over R math M otimes R N ... Tensor product of modules.png 200px right For every abelian group Z and every bilinear map math f ... properties , the above property defines the tensor product uniquely up to a unique isomorphism any ... see that the properties for the tensor product are satisfied. Therefore, the tensor product of Q and Z ... minimally so that this map is bilinear. The direct product of M and N is rarely isomorphic to the tensor product of M and N . When R is not commutative, then the tensor product requires that M and N ... the definition to a tensor product of any number of spaces. For example, the universal property ... sub M sub 2 sub M sub 3 sub . The tensor product of three modules defined by the universal property ... as described. In the construction of the tensor product over a commutative ring R , the multiplication ... ring, then their tensor product is again an R module. If R is a ring, sub R sub M is a left ... of R on M factors through an action of a quotient commutative ring. In this case the tensor product .... See also Tor functor Tensor product of algebras Tensor product of fields Notes references References ...   more details



  1. Tensor product of fields

    field N for example a prime field . The tensor product of fields is the best available construction ... their tensor product will in that case be the trivial ring collapse of the construction to nothing ... K . L as a vector space tensor product , taken over the field N that is the intersection of K and L ... ref The tensor product as ring To get a general theory, one needs to consider a ring structure on math K otimes mathrm N L math . One can define the product math a otimes b c otimes d math to be math ... on the tensor product, making math K otimes mathrm N L math into a commutative algebra over a field N algebra , called the tensor product of fields . Analysis of the ring structure The structure ... The kernel of will be a prime ideal of the tensor product and conversely any prime ideal of the tensor product will give a homomorphism of N algebras to an integral domain inside a field of fractions ... of the product of all embeddings of K and L in various M , over N . In case K and L are finite extensions of N, the situation is particularly simple since the tensor product is of finite dimension ... K otimes mathbf Q K math is the product of a copy of K , and a splitting field of X sup 3 sup 2, of degree 6 over Q . One can prove this by calculating the dimension of the tensor product over Q as 9 ..., of products of fields alone. See also Extension of scalars tensor product of a field extension and a module ...In abstract algebra , the theory of field mathematics fields lacks a direct product the direct product ... a direct product of fields it can, though, contain non zero nilpotents see radical of a ring . If K ... K otimes mathrm N mathrm L mathrm R math as a direct product of finitely many fields. Each such field ... theory of real and complex embeddings In algebraic number theory , tensor products of fields are implicitly ... R math is always a product of fields isomorphic to R or C . The totally real number field s are those ... where Q sub p sub is the field of p adic number p adic numbers . This is a product of finite extensions ...   more details



  1. Tensor product model transformation

    Orphan date May 2010 The tensor product TP model transformation originally proposed for control design by Baranyi ref name compind cite journal author P. Baranyi and D. Tikk and Y. Yam and R. J. Patton title From Differential Equations to PDC Controller Design via Numerical Transformation journal Computers in Industry, Elsevier Science year 2003 volume 51 pages 281&ndash 297 ref ref name Baranyi04 cite journal author P. Baranyi title TP model transformation as a way to LMI based controller design journal IEEE Transaction on Industrial Electronics year 2004 volume 51 number 2 month April pages 387&ndash 400 ref is capable of numerically reconstructing the higher order singular value decomposition HOSVD of continuous multi variable functions. The HOSVD of functions is defined by Baranyi et al. in ref name canon1 cite book author P. Baranyi and L. Szeidl and P. V rlaki and Y. Yam title Definition of the HOSVD based canonical form of polytopic dynamic models booktitle 3rd International Conference ... details could be found in ref name JGDC cite journal author P. Baranyi title Tensor Product Model Based ... of elements. The given function can be a matrix or tensor, and it is irrelevant whether it is given ... is based on the analogy of tensor HOSVD . The key difference is that instead of singular matrices ... of tensor HOSVD can be carried over to function HOSVD . In the same way the main component analysis ..., in the same manner as the tensor HOSVD is used for complexity reduction. The TP model transformation ... R N math . Note that math mathbf F mathbf x math can be matrix or even tensor. In the followings ... tensor notation math mathbf F mathbf x mathcal S boxtimes n 1 N mathbf w n x n , math where the N 2 dimensional coefficient tensor math mathcal S in mathcal R I 1 times I 2 times ldots I N times ... math are respectively assigned to the elements of the parameter vector math mathbf x math . For tensor ... S math stored in tensor math mathcal S math , for all math mathbf x in Omega math . The TP model transformation ...   more details



  1. Tensor

    is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor product ... is defined in this context as an element of the tensor product of vector spaces, ref Springer id a a011120 ... sub is a basis of V and w sub j sub is a basis of W , then the tensor product math V otimes W math has ... of the tensor product, it can be shown that these components satisfy the transformation law for a type m , n tensor. Moreover, the universal property of the tensor product gives a bijection 1 to 1 correspondence ... form is the same thing as a 0, 2 tensor an inner product is an example of a 0, 2 tensor, but not all ... functional , 1 form linear transformation m 2 bilinear form , e.g. inner product , metric tensor ..., however there also exist operations that change the type of the tensors. Tensor product Main Tensor product The tensor product takes two tensors, S and T , and produces a new tensor, S T , whose order ... n , m , then the tensor product S T has type k n , l m . Contraction Main Tensor contraction Tensor ... algebra trace . The contraction is often used in conjunction with the tensor product to contract ... as an element of a tensor product of copies of the space V with the space V sup sup by first ... learning Multilinear projection Multilinear projection One form Tensor product of modules Applications ...other uses Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . File Components stress tensor.svg right thumb 300px Stress, a second order tensor. The tensor s components ... examples of such relations include the dot product , the cross product , and linear map s. Vectors and scalars themselves are also tensors. A tensor can be represented as a Array data structure ... of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number ... by a matrix, a 2 dimensional array, and therefore is a 2nd order tensor. A vector can be represented as a 1 dimensional array and is a 1st order tensor. Scalars are single numbers and are thus zeroth ...   more details



  1. Tensor (disambiguation)

    Tensor Latin tensio tension , a tensor would then be someone who tenses may refer to Mathematics Tensor , geometric entities introduced into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices Antisymmetric tensor Bach tensor Completely antisymmetric tensor Cotton tensor Diffusion tensor Dyadic tensor Einstein tensor Electrogravitic tensor Electromagnetic stress energy tensor Electromagnetic tensor Four tensor Gyration tensor Inertia tensor Lanczos tensor Maxwell stress tensor Metric tensor general relativity Metric tensor Mixed tensor Polder tensor Ricci tensor Riemann curvature tensor Scalar tensor theory Schouten tensor Strain tensor Stress energy tensor Structure tensor Symmetric tensor Tensor intrinsic definition Tensor algebra Tensor bundle Tensor category Tensor contraction Tensor density Tensor field Tensor product of algebras Tensor product of modules Tensor product Topogravitic tensor Topological tensor product Torsion tensor Weyl tensor Anatomy Tensor veli palatini muscle Tensor fasciae latae muscle Tensor tympani muscle Other Tensor Trucks , skateboarding truck company See also Glossary of tensor theory Curvature tensor disambiguation Stress tensor disambiguation Tense disambiguation disambig ...   more details



  1. Schouten tensor

    In Riemannian geometry , the Schouten tensor is a second order tensor which is introduced by Jan Arnoldus Schouten . It is defined by, for n 3 dimensions, math P frac 1 n 2 left Ric frac R 2 n 1 g right , math where Ric is the Ricci tensor , R is the scalar curvature , g is the Riemannian metric and n is the dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni&ndash Nomizu product of the Schouten tensor with the metric. See also Weyl Schouten theorem Cotton tensor Category Riemannian geometry Category Tensors differential geometry stub ru ...   more details



  1. Weyl tensor

    n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor , s is the scalar curvature , and h small O small k denotes the Kulkarni&ndash Nomizu product ...In differential geometry , the Weyl curvature tensor , named after Hermann Weyl , is a measure of the curvature ... tensor , the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic . The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how ... force. The Ricci curvature , or trace linear algebra trace component of the Riemann tensor contains ... tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace free Tensor contraction Metric contraction ... tensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero. If the Weyl tensor ... system in which the metric tensor is proportional to a constant tensor. This fact was a key component ... . Definition The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a 0,4 valence tensor by contracting with the metric . The 0,4 valence Weyl tensor is then Harv Petersen 2006 p 92 NumBlk math W R ... 1,3 valent Weyl tensor is then given by contracting the above with the inverse of the metric. The decomposition EquationNote 1 expresses the Riemann tensor as an orthogonal direct sum of vector bundles ... decomposition , expresses the Riemann curvature tensor into its irreducible representation irreducible ... tensor further decomposes into invariant factors for the action of the special orthogonal group , the self dual and antiself dual parts W sup sup and W sup &minus sup . The Weyl tensor can also be expressed using the Schouten tensor , which is a trace adjusted multiple of the Ricci tensor, math ... math R abcd math is the Riemann tensor, math R ab math is the Ricci tensor, math R math is the Ricci ...   more details



  1. Tensor field

    the tensor product concept is independent of any choice of basis, taking the tensor product of two ... of W , the tensor product of V with L sup s sup , and consider tensor density fields with weight s . Half ...Unreferenced date August 2008 In mathematics , physics and engineering , a tensor field assigns a tensor to each point of a mathematical space typically a Euclidean space or manifold . Tensor fields are used ... physics stress and strain tensor strain in materials, and in numerous applications in the physical sciences and engineering. As a tensor is a generalization of a scalar physics scalar a pure number representing a value, like length and a Euclidean vector vector a geometrical arrow in space , a tensor ... tensor fields , that is, fields defined over a manifold which define a tensor at every point of the manifold. An example is the Riemann curvature tensor . Geometric introduction Intuitively, a vector ... velocity at each point of the Earth s surface. The general idea of tensor field combines the requirement ... tensor &mdash with the idea that we don t want our notion to depend on the particular method ... explanation The contemporary mathematical expression of the idea of tensor field breaks it down into a two ... of tensor field , namely as a section fiber bundle section of some tensor bundle . There are vector bundles which are not tensor bundles the M bius band for instance. This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space math V otimes cdots otimes V otimes ... . See also tangent bundle and cotangent bundle . Given two tensor bundles E M and F M , a map A E F from the space of sections of E to sections of F can be considered itself as a tensor section ..., where f is a smooth function on M . Thus a tensor is not only a linear map on the vector space ... tensor torsion and Affine connection curvature tensors built from them are. Notation The notation ...   more details



  1. Tensor density

    is 1. Multiplication of tensor densities A product of tensor densities of any types will have a weight equal to the sum of the weights of the factors. A product of authentic tensor densities ... densities. Similarly, a product of even tensor densities and odd tensor densities will be an even ...General relativity In differential geometry , a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system .... A distinction is made among authentic tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle . Definition Unreferenced section date June 2011 Some authors classify tensor densities into the two types called authentic tensor densities and pseudotensor ... tensor densities and odd tensor densities. When a tensor density weight is an integer there is an equivalence ... elucidate the different ways that tensor densities transform under the somewhat ... into these types, there is only one way that tensor densities transform under orientation preserving ... than 2 to the determinant of the metric tensor expressed with Covariance and contravariance of vectors covariant indices. cn date June 2011 Tensor and pseudotensor densities For example, a mixed rank two authentic tensor density of weight W transforms as math mathfrak T alpha beta left det left frac ... authentic tensor density of integer weight W where math bar mathfrak T math is the rank two tensor density in the math bar x math coordinate system, math mathfrak T math is the transformed tensor density ... is applicable only when W is an integer. However, see even and odd tensor densities below. We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation ... 1 when its argument is positive or 1 when its argument is negative. Even and odd tensor densities ...   more details



  1. Nonmetricity tensor

    unreferenced date March 2012 In mathematics , the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor . It is therefore a tensor field of Tensor Tensor rank rank three. It vanishes for the case of Riemannian geometry . Category Differential geometry Geometry stub ...   more details



  1. Tensor contraction

    from the Component free treatment of tensors Definition Tensor Product of Vector Spaces tensor product of these two spaces to the field k math C V otimes V rightarrow k math corresponding ... product also known as a Metric tensor metric g , such contractions are possible. One uses the metric ... product math T otimes U math is a new tensor, which, if it has at least one covariant and one contravariant ... the tensor product, yielding a composite tensor. Contracting two indices in this composite .... See also Tensor product Partial trace Interior product Raising and lowering indices Musical isomorphism ...In multilinear algebra , a tensor contraction is an operation on one or more tensor s that arises from ... vector space dual . In components, it is expressed as a sum of products of scalar components of the tensor ... to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices one a subscript, the other a superscript of the tensor are set equal to each other and summed ... tensor with Tensor Tensor rank rank or order reduced by 2. Tensor contraction can be seen as a generalization ... the contraction operation on a tensor of type 1,1 , which is an element of math V otimes V math ... obtains a basis free definition of the trace linear algebra trace . In general, a tensor of type m , n ... operation, which is a linear map which yields a tensor of type m   &minus   1,  n   ... dual basis . Since a general mixed dyadic tensor is a linear combination of decomposable tensors ... T T i j mathbf e i e j math be a mixed dyadic tensor. Then its contraction is math T i j mathbf ... being implied by the summation convention . The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type 2,2 on the second and third indices to create a new tensor U of type 1,1 is written as math T ab bc sum b T ab bc T a1 ... dyadic tensor. This tensor does not contract if its base vectors are dotted the result is the contravariant ...   more details



  1. Tensor algebra

    In mathematics , the tensor algebra of a vector space V , denoted T V or T sup sup V , is the algebra over a field algebra of tensor s on V of any rank with multiplication being the tensor product . It is the free ... the k sup th sup tensor power of V to be the tensor product of V with itself k times math T kV V ... math T kV otimes T ell V to T k ell V math given by the tensor product, which is then extended by linearity to all of T V . This multiplication rule implies that the tensor algebra T V is naturally a graded ... property see below . The tensor algebra also has two Coalgebra structures coalgebra structures ... of Tensor Tensor rank rank k . By convention T sup 0 sup V is the ground field K as a one dimensional ... in straightforward manner to the tensor algebra of any module mathematics module M over a commutative ... tensor products cannot be formed. Adjunction and universal property The tensor algebra T V ..., it s the functor which sends each K algebra to its underlying vector space. Explicitly, the tensor ... commutative diagram Image TensorAlgebra 01.png center Universal property of the tensor ..., define the tensor algebra T V as the unique algebra satisfying this property specifically, it is unique ... exists. The above universal property shows that the construction of the tensor algebra is functorial ... n , another way of looking at the tensor algebra is as the algebra of polynomials over K in n ... coordinate of the vector . Quotients Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain ... algebra s. Coalgebra structures The tensor algebra has two coalgebra structures one simple one, which ... structure on the tensor algebra is given as follows. The coproduct is defined by math Delta v 1 ... to bigoplus i j m T iV otimes T jV math and is also compatible with the grading. The tensor algebra ... . Finally, the tensor algebra becomes a Hopf algebra with antipode given by math S x 1 otimes ...   more details



  1. Symmetric tensor

    otimes r math a tensor of order r . Then T is a symmetric tensor if math tau sigma T T , math for the Tensor product Tensor powers and braiding braiding maps associated to every permutation &sigma on the symbols ...In mathematics , a symmetric tensor is tensor that is invariant under a permutation of its vector arguments. Thus an r sup th sup order symmetric tensor represented in coordinates as a quantity with r ... concept is that of the antisymmetric tensor or alternating form . Symmetric tensors occur ... . Given a basis linear algebra basis e sup i sup of V , any symmetric tensor T of rank r can ... e i r math for some unique list of coefficients math T i 1i 2 dots i r math the components of the tensor ... dim , operatorname Sym r V N r 1 choose r . math Symmetric part of a tensor Suppose math V math is a vector ... is a tensor of order math r math , then the symmetric part of math T math is the symmetric tensor ... r e i 1 otimes e i 2 otimes cdots otimes e i r . math The components of the tensor appearing on the right ... brackets are used to indicate anti symmetrization. If T is a simple tensor, given as a pure tensor product math T v 1 otimes v 2 otimes cdots otimes v r math then the symmetric part of T is the symmetric product of the factors math v 1 odot v 2 odot cdots odot v r frac 1 r sum sigma in mathfrak ... and field physics fields used in physics and engineering can be represented as symmetric tensor fields for example, stress physics stress , strain tensor strain , and anisotropic Electrical resistivity ... matrices , a real symmetric tensor of order 2 can be diagonalized . More precisely, for any tensor ... s law of inertia . The minimum number n for which such a decomposition is possible is the rank of a tensor rank of T . The vectors appearing in this minimal expression are the principal axes of the tensor ... tensor define the ellipsoid representing the moment of inertia. Ellipsoids are examples of algebraic ... tensor Ricci calculus Schur polynomial symmetric polynomial transpose Young symmetrizer Notes references ...   more details



  1. Stress tensor

    Stress tensor may refer to Stress mechanics , in classical physics Stress energy tensor , in relativistic theories Maxwell stress tensor , in electromagnetism See also Stress disambiguation Tensor disambiguation disambig de Spannungstensor he ...   more details



  1. Trifocal tensor

    In computer vision , the trifocal tensor also tritensor is a 3 3 3 array of numbers i.e., a tensor that incorporates all projective geometry projective geometric relationships among three views. It relates the coordinates of corresponding points or lines in three views, being independent of the scene structure and depending only on the relative motion i.e., pose among the three views and their intrinsic calibration parameters. Hence, the trifocal tensor can be considered as the generalization of the fundamental matrix computer vision fundamental matrix in three views. It is noted that despite that the tensor is made up of 27 elements, only 18 of them are actually independent. Correlation slices The tensor can also be seen as a collection of three rank two 3 x 3 matrices math mathbf T 1, mathbf T 2, mathbf T 3 math known as its correlation slices . Assuming that the Camera matrix projection matrices of three views are math mathbf P mathbf I mathbf 0 math , math mathbf P mathbf A mathbf a 4 math and math mathbf P mathbf B mathbf b 4 math , the correlation slices of the corresponding tensor can be expressed in closed form as math mathbf T i mathbf a i mathbf b 4 t mathbf a 4 mathbf b i t, i 1 ldots 3 math , where math mathbf a i, mathbf b i math are respectively the i sup th sup columns of the camera matrices. In practice, however, the tensor is estimated from point and line matches ... tensor is that it gives rise to linear relationships between lines and points in three images. More ... math denotes the skew symmetric Cross product Conversion to matrix multiplication cross product matrix . Transfer Given the trifocal tensor of three views and a pair of matched points in two views, it is possible ... University Press year 2003 isbn 0 521 54051 8 Chapter on tensor is online http www.robots.ox.ac.uk ... in Three Views and the Trifocal Tensor journal International Journal of Computer Vision volume 22 ... Zisserman title Robust Parameterization and Computation of the Trifocal Tensor journal Image and Vision ...   more details



  1. Dyadic tensor

    product associative law associates with the juxtaposition of vectors. The tensor contraction of a dyadic ... could possibly mean using the language of tensor products. If V is a finite dimensional vector space, a dyadic tensor on V is an elementary tensor in the tensor product of V with its dual space. 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 and do use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in Euclidean space. In this sense, the dyadic tensor i j ...In multilinear algebra , a dyadic is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, along with a notation for manipulating such expressions analogous to the rules for matrix mathematics matrix algebra . Each component of a dyadic is a dyad . A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient. As an example, let math mathbf A a mathbf ... partial x j A mn . math Thus a dyadic is a covariant tensor of order two. The dyadic itself, rather ... A can be combined with a vector v by means of the dot product math mathbf A cdot mathbf v sum ... of the dyadic in a coordinate basis by replacing each juxtaposition by a dot product of vectors ... mathbf e 2 A 31 A 13 mathbf e 3 A 12 A 21 math arises by replacing every juxtaposition by a cross product . The resulting vector is the complete contraction of A with the Levi Civita tensor math sum mn epsilon i mn A mn . math Examples The dyadic tensor J j i &minus i j math left begin array cc 0 & 1 1 & 0 end array right math is a 90 rotation operator in two dimensions. It can be dot product dotted ... dyadic tensor in three dimensions is I i i j j k k i sup T sup i j sup T sup j k sup T sup k . This can ... in that basis. See also Dyadic product Dyadics Notes references References Citation last1 Morse first1 ...   more details



  1. Topogravitic tensor

    Unreferenced stub auto yes date December 2009 In general relativity , the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor . The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame fields in general relativity frame field . DEFAULTSORT Topogravitic Tensor Category Tensors in general relativity Relativity stub ...   more details



  1. Electromagnetic tensor

    Inner product and Lorentz covariance If one forms an inner product of the field strength tensor a Lorentz ... tensor In electromagnetism , electromagnetic tensor or electromagnetic field tensor sometimes called the field strength tensor , Faraday tensor or Maxwell bivector is a mathematical object that describes the electromagnetic field of a physical system. The field tensor was first used after the 4 dimensional tensor formulation of special relativity was introduced by Hermann Minkowski . The tensor allows some physical laws to be written in a very concise form. Definition The electromagnetic tensor ... A , math By definition, the electromagnetic tensor is the Exterior derivative Exterior derivative of a k ... form of the field tensor yields the following properties ref cite book author J.A. Wheeler, C. Misner ... does not change from one frame of reference to another. li Pseudoscalar invariant The product of the tensor math F mu nu , math with its dual tensor gives the invariant math frac 1 2 epsilon alpha ... gamma alpha 0 math or using the Antisymmetric tensor index notation with square brackets ref antisymmetric note 1 for the antisymmetric part of the tensor math partial alpha F beta gamma 0 math Relativity main Maxwell s equations in curved spacetime The field tensor derives its name from the fact that the electromagnetic field is found to obey the tensor transformation law , this general property ... systems this led to the introduction of tensor s. The tensor formalism also leads to a mathematically ... the field tensor, so this finally simplifies to math partial nu F mu nu 0 math That equation ... . In quantum field theory it is used as the template for the gauge field strength tensor. By being ... calculus Application of tensor theory in physics Classification of electromagnetic fields Covariant ... Category Tensors in general relativity ca Tensor electromagn tic de Elektromagnetischer Feldst rketensor et Elektromagnetv lja tensor es Tensor de campo electromagn tico fr Tenseur lectromagn tique ...   more details




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