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Concept
Outer product
Summary
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
Standard matrix multiplication
Given two vectors of size and respectively
their outer product, denoted is defined as the matrix obtained by multiplying each element of by each element of :
Or in index notation:
Denoting the dot product by if given an vector then If given a vector then
If and are vectors of the same dimension bigger than 1, then .
The outer product is equivalent to a matrix multiplication provided that is represented as a column vector and as a column vector (which makes a row vector). For instance, if and then
For complex vectors, it is often useful to take the conjugate transpose of denoted or :
If then one can take the matrix product the other way, yielding a scalar (or matrix):
which is the standard inner product for Euclidean vector spaces, better known as the dot product. The dot product is the trace of the outer product. Unlike the dot product, the outer product is not commutative.
Multiplication of a vector by the matrix can be written in terms of the inner product, using the relation .
Given two tensors with dimensions and , their outer product is a tensor with dimensions and entries
For example, if is of order 3 with dimensions and is of order 2 with dimensions then their outer product is of order 5 with dimensions If has a component A[2, 2, 4] = 11 and has a component B[8, 88] = 13, then the component of formed by the outer product is C[2, 2, 4, 8, 88] = 143.
Official source
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