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

Definition

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

$${\displaystyle \mathbf {u} ={\begin{bmatrix}u_{1}\\u_{2}\\\vdots \\u_{m}\end{bmatrix}},\quad \mathbf {v} ={\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}}$$

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 ${\displaystyle \mathbf {v} }$:^[1]

$${\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {A} ={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&\dots &u_{1}v_{n}\\u_{2}v_{1}&u_{2}v_{2}&\dots &u_{2}v_{n}\\\vdots &\vdots &\ddots &\vdots \\u_{m}v_{1}&u_{m}v_{2}&\dots &u_{m}v_{n}\end{bmatrix}}}$$

Or, in index notation:

$${\displaystyle (\mathbf {u} \otimes \mathbf {v} )_{ij}=u_{i}v_{j}}$$

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).^[2][3] For instance, if ${\displaystyle m=4}$ and ${\displaystyle n=3,}$ then^[4]

$${\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\textsf {T}}={\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\\u_{4}\end{bmatrix}}{\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&u_{1}v_{3}\\u_{2}v_{1}&u_{2}v_{2}&u_{2}v_{3}\\u_{3}v_{1}&u_{3}v_{2}&u_{3}v_{3}\\u_{4}v_{1}&u_{4}v_{2}&u_{4}v_{3}\end{bmatrix}}.}$$

For complex vectors, it is often useful to take the conjugate transpose of ${\displaystyle \mathbf {v} ,}$ denoted ${\displaystyle \mathbf {v} ^{\dagger }}$ or ${\displaystyle \left(\mathbf {v} ^{\textsf {T}}\right)^{*}}$:

$${\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\dagger }=\mathbf {u} \left(\mathbf {v} ^{\textsf {T}}\right)^{*}.}$$

Properties

The outer product of vectors satisfies the following properties:

$${\displaystyle {\begin{aligned}(\mathbf {u} \otimes \mathbf {v} )^{\textsf {T}}&=(\mathbf {v} \otimes \mathbf {u} )\\(\mathbf {v} +\mathbf {w} )\otimes \mathbf {u} &=\mathbf {v} \otimes \mathbf {u} +\mathbf {w} \otimes \mathbf {u} \\\mathbf {u} \otimes (\mathbf {v} +\mathbf {w} )&=\mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {w} \\c(\mathbf {v} \otimes \mathbf {u} )&=(c\mathbf {v} )\otimes \mathbf {u} =\mathbf {v} \otimes (c\mathbf {u} )\end{aligned}}}$$

The outer product of tensors satisfies the additional associativity property:

$${\displaystyle (\mathbf {u} \otimes \mathbf {v} )\otimes \mathbf {w} =\mathbf {u} \otimes (\mathbf {v} \otimes \mathbf {w} )}$$

Definition (abstract)

Let V and W be two vector spaces. The outer product of ${\displaystyle \mathbf {v} \in V}$ and ${\displaystyle \mathbf {w} \in W}$ is the element ${\displaystyle \mathbf {v} \otimes \mathbf {w} \in V\otimes W}$.

If V is an inner product space, then it is possible to define the outer product as a linear map V → W. In this case, the linear map ${\displaystyle \mathbf {x} \mapsto \langle \mathbf {v} ,\mathbf {x} \rangle }$ is an element of the dual space of V, as this maps linearly a vector into its underlying field, of which ${\displaystyle \langle \mathbf {v} ,\mathbf {x} \rangle }$ is an element. The outer product V → W is then given by

$${\displaystyle (\mathbf {w} \otimes \mathbf {v} )(\mathbf {x} )=\left\langle \mathbf {v} ,\mathbf {x} \right\rangle \mathbf {w} .}$$

This shows why a conjugate transpose of v is commonly taken in the complex case.

In programming languages

In some programming languages, given a two-argument function f (or a binary operator), the outer product, f, of two one-dimensional arrays, A and B, is a two-dimensional array C such that C[i, j] = f(A[i], B[j]). This is syntactically represented in various ways: in APL, as the infix binary operator ∘.f; in J, as the postfix adverb f/; in R, as the function outer(A, B, f) or the special %o%;^[7] in Mathematica, as Outer[f, A, B]. In MATLAB, the function kron(A, B) is used for this product. These often generalize to multi-dimensional arguments, and more than two arguments.

In the Python library NumPy, the outer product can be computed with function np.outer().^[8] In contrast, np.kron results in a flat array. The outer product of multidimensional arrays can be computed using np.multiply.outer.

Applications

As the outer product is closely related to the Kronecker product, some of the applications of the Kronecker product use outer products. These applications are found in quantum theory, signal processing, and image compression.^[9]

See also

• Dyadics
• Householder transformation
• Norm (mathematics)
• Ricci calculus
• Scatter matrix

References

Further reading

• Carlen, Eric; Canceicao Carvalho, Maria (2006). "Outer Products and Orthogonal Projections". Linear Algebra: From the Beginning. Macmillan. pp. 217–218. ISBN 9780716748946.