Gamas's Theorem
Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group to be zero. It
Multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function where and are vector spaces (or modu
Multilinear form
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map that is separately K-linear in each of its k arguments. More generally, one can define multiline
Cryptographic multilinear map
A cryptographic -multilinear map is a kind of multilinear map, that is, a function such that for any integers and elements , , and which in addition is efficiently computable and satisfies some securi
Homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, is a homogeneous polynomial of degree 5, i
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vect
Plücker coordinates
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, P3. Because they satisfy a quadr
Higher-order singular value decomposition
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular
Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an ex
Tensor product of modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is an
Mode-k flattening
In multi-linear algebra, mode-k flattening (also matricisation, matricizing, or unfolding) is an operation on tensor (a multi-dimensional array) denoted by turning it into a matrix (a two-dimensional
Hyperdeterminant
In algebra, the hyperdeterminant is a generalization of the determinant. Whereas a determinant is a scalar valued function defined on an n × n square matrix, a hyperdeterminant is defined on a multidi
Berezin integral
In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variabl
Interior product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)
Alternating multilinear map
In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a m
Multilinear multiplication
In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
Paravector
The name paravector is used for the sum of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists. This name was given by J. G. Maks in a doctoral dissertation at T
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, i
Grassmann–Cayley algebra
In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in wh
Cubic form
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic
Tensor rank decomposition
In multilinear algebra, the tensor rank decomposition, the exact decomposition of a tensor in terms of the minimum terms, is an open problem. Canonical polyadic decomposition (CPD) is a variant of the
Tensor reshaping
In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order- tensor and the set of indices of an order- tensor, where . The use of indices presupposes tensor
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consis
Skew lines
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular te
Tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of s
Exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exte
Bilinear form
In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a
Trace diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled
Pfaffian
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the siz
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational conve
Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic
Binet–Cauchy identity
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that for every choice of real or complex numbers (or more generally, elements of a com
Symmetric algebra
In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. H
Discrete exterior calculus
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. DEC methods have proved to be very powerfu
Glossary of tensor theory
This is a glossary of tensor theory. For expositions of from different points of view, see:
* Tensor
* Tensor (intrinsic definition)
* Application of tensor theory in engineering science For some h
Lagrange's identity
In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is: which applies to any two sets {a1, a2, ..., an} and {b1, b2, ..., bn} of real or complex numbers (or more generally, elements of
Multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multil
Multilinear subspace learning
Multilinear subspace learning is an approach to dimensionality reduction. Dimensionality reduction can be performed on a data tensor whose observations have been vectorized and organized into a data t
Triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scal