Outer product

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 $$m \times 1$$ and $$n \times 1$$ respectively
 * $$\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}$$ their outer product, denoted $$\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 $\mathbf{v}$:

\mathbf{u} \otimes \mathbf{v} = \mathbf{A} = \begin{bmatrix} u_1v_1 & u_1v_2 & \dots & u_1v_n \\ u_2v_1 & u_2v_2 & \dots & u_2v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_mv_1 & u_mv_2 & \dots & u_mv_n \end{bmatrix} $$

Or, in index notation:
 * $$(\mathbf{u} \otimes \mathbf{v})_{ij} = u_i v_j$$

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

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

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

\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 $$\mathbf{v},$$ denoted $$\mathbf{v}^\dagger$$ or $$\left(\mathbf{v}^\textsf{T}\right)^*$$:
 * $$\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\dagger = \mathbf{u} \left(\mathbf{v}^\textsf{T}\right)^*.$$

Contrast with Euclidean inner product
If $$m = n,$$ then one can take the matrix product the other way, yielding a scalar (or $$1 \times 1$$ matrix):
 * $$\left\langle\mathbf{u}, \mathbf{v}\right\rangle = \mathbf{u}^\textsf{T} \mathbf{v}$$

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 $$\mathbf{w}$$ by the matrix $$\mathbf{u} \otimes \mathbf{v}$$ can be written in terms of the inner product, using the relation $$\left(\mathbf{u} \otimes \mathbf{v}\right)\mathbf{w} = \mathbf{u}\left\langle\mathbf{v}, \mathbf{w}\right\rangle$$.

The outer product of tensors
Given two tensors $$\mathbf{u}, \mathbf{v}$$ with dimensions $$(k_1, k_2, \dots, k_m)$$ and $$(l_1, l_2, \dots, l_n)$$, their outer product $$\mathbf{u} \otimes \mathbf{v}$$ is a tensor with dimensions $$(k_1, k_2, \dots, k_m, l_1, l_2, \dots, l_n)$$ and entries
 * $$(\mathbf{u} \otimes \mathbf{v})_{i_1, i_2, \dots i_m, j_1, j_2, \dots, j_n} = u_{i_1, i_2, \dots, i_m} v_{j_1, j_2, \dots, j_n}$$

For example, if $$\mathbf{A}$$ is of order 3 with dimensions $$(3, 5, 7)$$ and $$\mathbf{B}$$ is of order 2 with dimensions $$(10, 100),$$ then their outer product $$\mathbf{C}$$ is of order 5 with dimensions $$(3, 5, 7, 10, 100).$$ If $$\mathbf{A}$$ has a component $A_{[2, 2, 4]} = 11$ and $$\mathbf{B}$$ has a component $B_{[8, 88]} = 13$, then the component of $$\mathbf{C}$$ formed by the outer product is $C_{[2, 2, 4, 8, 88]} = 143$.

Connection with the Kronecker product
The outer product and Kronecker product are closely related; in fact the same symbol is commonly used to denote both operations.

If $$\mathbf{u} = \begin{bmatrix}1 & 2 & 3\end{bmatrix}^\textsf{T}$$ and $$\mathbf{v} = \begin{bmatrix}4 & 5\end{bmatrix}^\textsf{T}$$, we have:
 * $$\begin{align}

\mathbf{u} \otimes_\text{Kron} \mathbf{v} &= \begin{bmatrix} 4 \\ 5 \\ 8 \\ 10 \\ 12 \\ 15\end{bmatrix}, & \mathbf{u} \otimes_\text{outer} \mathbf{v} &= \begin{bmatrix} 4 & 5 \\ 8 & 10 \\ 12 & 15\end{bmatrix} \end{align}$$

In the case of column vectors, the Kronecker product can be viewed as a form of vectorization (or flattening) of the outer product. In particular, for two column vectors $$\mathbf{u}$$ and $$\mathbf{v}$$, we can write:
 * $$\mathbf{u} \otimes_{\text{Kron}} \mathbf{v} = \operatorname{vec}(\mathbf{v} \otimes_\text{outer} \mathbf{u})$$

(The order of the vectors is reversed on the right side of the equation.)

Another similar identity that further highlights the similarity between the operations is
 * $$\mathbf{u} \otimes_{\text{Kron}} \mathbf{v}^\textsf{T} = \mathbf u \mathbf{v}^\textsf{T} = \mathbf{u} \otimes_{\text{outer}} \mathbf{v}$$

where the order of vectors needs not be flipped. The middle expression uses matrix multiplication, where the vectors are considered as column/row matrices.

Connection with the matrix product
Given a pair of matrices $$\mathbf{A}$$ of size $$m\times p$$ and $$\mathbf{B}$$ of size $$p\times n$$, consider the matrix product $$\mathbf{C} = \mathbf{A}\,\mathbf{B}$$ defined as usual as a matrix of size $$m\times n$$.

Now let $$\mathbf a^\text{col}_k$$ be the $$k$$-th column vector of $$\mathbf A$$ and let $$\mathbf b^\text{row}_k$$ be the $$k$$-th row vector of $$\mathbf B$$. Then $$\mathbf{C}$$ can be expressed as a sum of column-by-row outer products:
 * $$\mathbf{C} = \mathbf{A}\, \mathbf{B} =

\left(     \sum_{k=1}^p {A}_{ik}\, {B}_{kj} \right)_{ \begin{matrix} 1\le i \le m \\[-20pt] 1 \le j\le n \end{matrix} } = \begin{bmatrix} & & \\ \mathbf a^\text{col}_{1} & \cdots & \mathbf a^\text{col}_{p} \\ & & \end{bmatrix} \begin{bmatrix} & \mathbf b^\text{row}_{1} & \\ & \vdots & \\ & \mathbf b^\text{row}_{p} & \end{bmatrix} = \sum_{k=1}^p \mathbf a^\text{col}_k \otimes \mathbf b^\text{row}_k$$ This expression has duality with the more common one as a matrix built with row-by-column inner product entries (or dot product): $$C_{ij} = \langle{\mathbf a^\text{row}_i,\,\mathbf b_j^\text{col}}\rangle$$

This relation is relevant in the application of the Singular Value Decomposition (SVD) (and Spectral Decomposition as a special case). In particular, the decomposition can be interpreted as the sum of outer products of each left ($$\mathbf{u}_k$$) and right ($$\mathbf{v}_k$$) singular vectors, scaled by the corresponding nonzero singular value $$\sigma_k$$:
 * $$\mathbf{A} = \mathbf{U \Sigma V^T} = \sum_{k=1}^{\operatorname{rank}(A)}(\mathbf{u}_k \otimes \mathbf{v}_k) \, \sigma_k$$

This result implies that $$\mathbf{A}$$ can be expressed as a sum of rank-1 matrices with spectral norm $$\sigma_k$$ in decreasing order. This explains the fact why, in general, the last terms contribute less, which motivates the use of the truncated SVD as an approximation. The first term is the least squares fit of a matrix to an outer product of vectors.

Properties
The outer product of vectors satisfies the following properties:
 * $$\begin{align}

(\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{align}$$

The outer product of tensors satisfies the additional associativity property:

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

Rank of an outer product
If u and v are both nonzero, then the outer product matrix uvT always has matrix rank 1. Indeed, the columns of the outer product are all proportional to the first column. Thus they are all linearly dependent on that one column, hence the matrix is of rank one.

("Matrix rank" should not be confused with "tensor order", or "tensor degree", which is sometimes referred to as "rank".)

Definition (abstract)
Let $V$ and $W$ be two vector spaces. The outer product of $$\mathbf v \in V$$ and $$\mathbf w \in W$$ is the element $$\mathbf v \otimes \mathbf w \in V \otimes W$$.

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 $$\mathbf x \mapsto \langle \mathbf v, \mathbf x\rangle$$ is an element of the dual space of $V$, as this maps linearly a vector into its underlying field, of which $$\langle \mathbf v, \mathbf x\rangle$$ is an element. The outer product $V → W$ is then given by
 * $$(\mathbf w \otimes \mathbf v) (\mathbf x) = \left\langle \mathbf v, \mathbf x \right\rangle \mathbf w.$$

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  (or a binary operator), the outer product, , of two one-dimensional arrays,   and  , is a two-dimensional array   such that. This is syntactically represented in various ways: in APL, as the infix binary operator ; in J, as the postfix adverb  ; in R, as the function   or the special  ; in Mathematica, as. In MATLAB, the function  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. In contrast,  results in a flat array. The outer product of multidimensional arrays can be computed using.

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.

Spinors
Suppose $s, t, w, z ∈ C$ so that $(s, t)$ and $(w, z)$ are in $C^{2}$. Then the outer product of these complex 2-vectors is an element of $M(2, C)$, the 2 × 2 complex matrices:
 * $$\begin{pmatrix} sw & tw \\ sz & tz \end{pmatrix}.$$

The determinant of this matrix is $swtz − sztw = 0$ because of the commutative property of $C$.

In the theory of spinors in three dimensions, these matrices are associated with isotropic vectors due to this null property. Élie Cartan described this construction in 1937, but it was introduced by Wolfgang Pauli in 1927 so that $M(2,C)$ has come to be called Pauli algebra.

Concepts
The block form of outer products is useful in classification. Concept analysis is a study that depends on certain outer products:

When a vector has only zeros and ones as entries, it is called a logical vector, a special case of a logical matrix. The logical operation and takes the place of multiplication. The outer product of two logical vectors $(u_{i})$ and $(v_{j})$ is given by the logical matrix $$\left(a_{ij}\right) = \left(u_i \land v_j\right)$$. This type of matrix is used in the study of binary relations, and is called a rectangular relation or a cross-vector.

Products

 * Cartesian product
 * Cross product
 * Exterior product
 * Hadamard product (matrices)

Duality

 * Bra–ket notation for outer product
 * Complex conjugate
 * Conjugate transpose
 * Transpose