In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second 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.
The outer product contrasts with:
The dot product (a special case of "inner product"), which takes a pair of coordinate vectors as input and produces a scalar
The Kronecker product, which takes a pair of matrices as input and produces a block matrix
In linear algebra, the outerproduct of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element...
"inner product" is opposed to outerproduct, which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a 1...
then the outerproduct between x {\displaystyle x} and y {\displaystyle y} is defined as x y T {\displaystyle xy^{T}} . The Kronecker product is related...
multiplication Metric tensor Multiplication of vectors Outerproduct The term scalar product means literally "product with a scalar as a result". It is also used...
normal subgroup. an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular...
product satisfies (see below). If arranged into a rectangular array, the coordinate vector of x ⊗ y {\displaystyle x\otimes y} is the outerproduct of...
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation...
of the diagonal elements, hence the sum over a common index Aii. The outerproduct of the column vector ui by the row vector vj yields an m × n matrix...
Kronecker product or tensor product, the generalization to any size of the preceding Khatri-Rao product and Face-splitting productOuterproduct, also called...
called the exterior product (frequently called the wedge product and less often the outerproduct). It is standard to denote these respectively by juxtaposition...
{\displaystyle \mathbf {A} } multiplied by its magnitude. Specifically, for the outerproduct of two vectors, ∇ ⋅ ( A B T ) = B ( ∇ ⋅ A ) + ( A ⋅ ∇ ) B . {\displaystyle...
where ( a ⊙ b ) i = a i b i {\displaystyle (a\odot b)_{i}=a_{i}b_{i}} . Outerproduct - where ( a ⊗ b ) {\displaystyle (\mathbf {a} \otimes \mathbf {b} )}...
row and column vectors, a 3×3 matrix (also the result of the outerproduct or tensor product of a and b): a b ≡ a ⊗ b ≡ a b T = ( a 1 a 2 a 3 ) ( b 1 b...
is the 3 × 3 identity matrix and ⊗ {\displaystyle \otimes } is the outerproduct. Further generalization of the parallel axis theorem gives the inertia...
{\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}} is the outerproduct matrix formed from the unit vector k ^ {\displaystyle \mathbf {\hat...
covariance matrix Wishart distribution Outerproduct—XX⊤{\displaystyle XX^{\top }}or X⊗X is the outerproduct of X with itself. Gram matrix Raghavan (2018-08-16)...
linear equations, higher-dimensional spaces, determinants, inner and outerproducts, and dual spaces. It is a mathematical tool used in engineering, machine...
In the fantasy role-playing game Dungeons & Dragons, an Outer Plane is one of a number of general types of planes of existence. They can also be referred...
(signifying null) then the composite function is an outerproduct, otherwise it is an inner product. An inner product intended for conventional matrix multiplication...
_{i}^{*}\end{pmatrix}}.} Note that bi b*i is an outerproduct, therefore this algorithm is called the outer-product version in (Golub & Van Loan). This is repeated...
matrix transposition. Load/store/insert/extract tile vectors. Matrix outerproduct of SVE vectors. "Streaming mode" SVE. Enhanced support for PCIe hot...
of the program. In APL it is also used for generalised inner product and outerproduct. In Erlang, Prolog, and Smalltalk, it marks the end of a statement...
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