Outer product

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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.

Contents

Definition

Given a vector \mathbf{u}=(u_1, u_2, \dots, u_m) with m elements and a vector \mathbf{v}= (v_1, v_2, \dots, v_n) with n elements, their outer product \mathbf{u} \otimes \mathbf{v} is defined as the m\times n matrix \mathbf{A} obtained by multiplying each element of \mathbf{u} by each element of \mathbf{v}:

Note that \mathbf{A} \mathbf{v} = \mathbf{u} \Vert v \Vert ^2.

For complex vectors, it is customary to use the complex conjugate of \mathbf{v} (denoted \bar \mathbf{v}). Namely, matrix \mathbf{A} is obtained by multiplying each element of \mathbf{u} by the complex conjugate of each element of \mathbf{v}.

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