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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