# Outer Product

In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix. The name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar.

The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices.

Some authors use the expression "outer product of tensors" as a synonym of "tensor product". The outer product is also a higher-order function in some computer programming languages such as APL and Mathematica.

### Other articles related to "outer product, product":

Bra-ket Notation - Linear Operators - Outer Products
... A convenient way to define linear operators on H is given by the outer product if is a bra and is a ket, the outer product denotes the rank-one operator that ... For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication The outer product is an N×N matrix, as expected for a ... One of the uses of the outer product is to construct projection operators ...
Matrix Multiplication - The Inner and Outer Products
... Given two column vectors a and b, the Euclidean inner product and outer product are the simplest special cases of the matrix product, by transposing the column vectors into row vectors ... The inner product is a column vector multiplied on the left by a row vector More explicitly, The outer product is a row vector multiplied on the left by a column vector where Matrix product (in terms of inner ... Matrix product (in terms of outer product) An alternative method results when the decomposition is done the other way around, i.e ...
Outer Product - Applications
... The outer product is useful in computing physical quantities (e.g ... the tensor of inertia), and performing transform operations in digital signal processing and digital image processing ...
APL Syntax And Symbols - Operators
... (first axis) ⍀ +⍀B Running sum down B U+2340 Inner product ... A+.×B Matrix product of A and B U+002E Outer product ∘ ... A∘.×B Outer product of A and B U+2218, U+002E The reduce and scan operators expect a dyadic function on their left, forming a monadic composite function applied to the vector on its right ...

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