In cosmological perturbation theory, the scalar–vector–tensor decomposition is a decomposition of the most general linearized perturbations of the...

} . Scalar–vector–tensor decomposition Hodge theory generalizing Helmholtz decomposition Polar factorization theorem Helmholtz–Leray decomposition used...

tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factorizes data tensors into smaller tensors....

such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual...

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...

of two vectors. A quaternion can be represented as the sum of a scalar and a vector. It can also be represented as the product of its tensor and its...

(known as scalar bosons), such as the Higgs field. These fields are the subject of scalar field theory. A vector field is an assignment of a vector to each...

states, and operators or tensor trains; Online Tensor Decompositions hierarchical Tucker decomposition; block term decomposition This section introduces...

with scalar fields, and 1-forms and 2-forms with vector fields: grad takes a scalar field (0-form) to a vector field (1-form); curl takes a vector field...

relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential...

element of which is termed a tensor. A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form: V ⊗...

m\times n} ⁠ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an m × n {\displaystyle m\times n} complex...

mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...

must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed...

the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two...

contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found...

Riemann 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...

as an antisymmetric rank-2 tensor. The addition of angular velocity vectors for frames is also defined by the usual vector addition (composition of linear...

and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced...

curvature tensor. Alternatively, in a coordinate-free notation one may use Riem for the Riemann tensor, Ric for the Ricci tensor and R for the scalar curvature...

bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero...

preserves the metric tensor. Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous...

the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle...

nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ...

to any vector space Levi-Civita symbol – Antisymmetric permutation object acting on tensors Ricci calculus – Tensor index notation for tensor-based calculations...

{K} } with a given decomposition of subspaces of grade 0 {\displaystyle 0} and grade 1 {\displaystyle 1} . The study of super vector spaces and their generalizations...

In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued...

Z{\Big )}.} With this convention, the Ricci tensor is a (0,2)-tensor field defined by Rjk=gilRijkl and the scalar curvature is defined by R=gjkRjk. (Note...

multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal...

algebraic tensor manipulation. Tensor is an R package for basic tensor operations. rTensor provides several tensor decomposition approaches. nnTensor provides...