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:

Given two vectors of size ${\displaystyle m\times 1}$ and ${\displaystyle n\times 1}$ respectively

their outer product, denoted ${\displaystyle \mathbf {u} \otimes \mathbf {v} ,}$ is defined as the ${\displaystyle m\times n}$ matrix ${\displaystyle \mathbf {A} }$ obtained by multiplying each element of ${\displaystyle \mathbf {u} }$ by each element of ${\displaystyle \mathbf {v} }$:

Or, in index notation:

Denoting the dot product by ${\displaystyle \,\cdot ,\,}$ if given an ${\displaystyle n\times 1}$ vector ${\displaystyle \mathbf {w} ,}$ then ${\displaystyle (\mathbf {u} \otimes \mathbf {v} )\mathbf {w} =(\mathbf {v} \cdot \mathbf {w} )\mathbf {u} .}$ If given a ${\displaystyle 1\times m}$ vector ${\displaystyle \mathbf {x} ,}$ then ${\displaystyle \mathbf {x} (\mathbf {u} \otimes \mathbf {v} )=(\mathbf {x} \cdot \mathbf {u} )\mathbf {v} ^{\operatorname {T} }.}$

If ${\displaystyle \mathbf {u} }$ and ${\displaystyle \mathbf {v} }$ are vectors of the same dimension bigger than 1, then ${\displaystyle \det(\mathbf {u} \otimes \mathbf {v} )=0}$.

The outer product ${\displaystyle \mathbf {u} \otimes \mathbf {v} }$ is equivalent to a matrix multiplication ${\displaystyle \mathbf {u} \mathbf {v} ^{\operatorname {T} },}$ provided that ${\displaystyle \mathbf {u} }$ is represented as a ${\displaystyle m\times 1}$ column vector and ${\displaystyle \mathbf {v} }$ as a ${\displaystyle n\times 1}$ column vector (which makes ${\displaystyle \mathbf {v} ^{\operatorname {T} }}$ a row vector). For instance, if ${\displaystyle m=4}$ and ${\displaystyle n=3,}$ then