Editing Outer product

{{short description|Vector operation}}
{{distinguish|text=[[Exterior product]]}}
In [[linear algebra]], the '''outer product''' of two [[coordinate vector]]s 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 [[tensor]]s (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 (mathematics)|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 <math>m \times 1</math> and <math>n \times 1</math> respectively
:<math display="block">\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}</math>
their outer product, denoted <math>\mathbf{u} \otimes \mathbf{v},</math> is defined as the <math>m \times n</math> matrix <math>\mathbf{A}</math> obtained by multiplying each element of <math>\mathbf{u}</math> by each element of {{nowrap|<math>\mathbf{v}</math>:}}<ref>{{cite book |title=Encyclopaedia of Physics |edition=2nd |first1=R. G. |last1=Lerner |author1-link=Rita G. Lerner |first2=G. L. |last2=Trigg |publisher=VHC |year=1991 |isbn=0-89573-752-3 |url-access=registration |url=https://archive.org/details/encyclopediaofph00lern}}</ref>
:<math display="block">
  \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}
</math>

Or, in index notation:
:<math display="block">(\mathbf{u} \otimes \mathbf{v})_{ij} = u_i v_j</math>

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

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

The outer product <math>\mathbf{u} \otimes \mathbf{v}</math> is equivalent to a [[matrix multiplication]] <math>\mathbf{u} \mathbf{v}^{\operatorname{T}},</math> provided that <math>\mathbf{u}</math> is represented as a <math>m \times 1</math> [[column vector]] and <math>\mathbf{v}</math> as a <math>n \times 1</math> column vector (which makes <math>\mathbf{v}^{\operatorname{T}}</math> a row vector).<ref>{{cite book |title=Linear Algebra |edition=4th |first1=S. |last1=Lipschutz |first2=M. |last2=Lipson |series=Schaum’s Outlines |publisher=McGraw-Hill |year=2009 |isbn=978-0-07-154352-1}}</ref><ref name=":0">{{cite web |last=Keller |first=Frank |date=February 23, 2020 |title=Algebraic Properties of Matrices; Transpose; Inner and Outer Product |url=https://www.inf.ed.ac.uk/teaching/courses/cfcs1/lectures/cfcs_l10.pdf |archive-url=https://web.archive.org/web/20171215061654/http://www.inf.ed.ac.uk/teaching/courses/cfcs1/lectures/cfcs_l10.pdf |archive-date=2017-12-15 |url-status=live |access-date=September 6, 2020 |website=inf.ed.ac.uk}}</ref> For instance, if <math>m = 4</math> and <math>n = 3,</math> then<ref>James M. Ortega (1987) ''Matrix Theory: A Second Course'', page 7, [[Plenum Press]] {{ISBN|0-306-42433-9}}</ref>
:<math display="block">
  \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}.
</math>

For [[complex numbers|complex]] vectors, it is often useful to take the [[conjugate transpose]] of <math>\mathbf{v},</math> denoted <math>\mathbf{v}^\dagger</math> or <math>\left(\mathbf{v}^\textsf{T}\right)^*</math>:
:<math display="block">\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\dagger = \mathbf{u} \left(\mathbf{v}^\textsf{T}\right)^*.</math>

===Contrast with Euclidean inner product===
If <math>m = n,</math> then one can take the matrix product the other way, yielding a scalar (or <math>1 \times 1</math> matrix):
:<math display="block">\left\langle\mathbf{u}, \mathbf{v}\right\rangle = \mathbf{u}^\textsf{T} \mathbf{v}</math>
which is the standard [[inner product]] for [[Euclidean vector space]]s,<ref name=":0"/> better known as the [[dot product]]. The dot product is the [[trace (linear algebra)|trace]] of the outer product.<ref>{{cite book |first=Robert F. |last=Stengel |title=Optimal Control and Estimation |location=New York |publisher=Dover Publications |year=1994 |page=26 |isbn=0-486-68200-5 |url=https://books.google.com/books?id=jDjPxqm7Lw0C&pg=PA26}}</ref> Unlike the dot product, the outer product is not commutative.

Multiplication of a vector <math>\mathbf{w}</math> by the matrix <math>\mathbf{u} \otimes \mathbf{v}</math> can be written in terms of the inner product, using the relation <math>\left(\mathbf{u} \otimes \mathbf{v}\right)\mathbf{w} = \mathbf{u}\left\langle\mathbf{v}, \mathbf{w}\right\rangle</math>.

===The outer product of tensors===
Given two tensors <math>\mathbf{u}, \mathbf{v}</math> with dimensions <math>(k_1, k_2, \dots, k_m)</math> and <math>(l_1, l_2, \dots, l_n)</math>, their outer product <math>\mathbf{u} \otimes \mathbf{v}</math> is a tensor with dimensions <math>(k_1, k_2, \dots, k_m, l_1, l_2, \dots, l_n)</math> and entries
:<math display="block">(\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}</math>

For example, if <math>\mathbf{A}</math> is of order 3 with dimensions <math>(3, 5, 7)</math> and <math>\mathbf{B}</math> is of order 2 with dimensions <math>(10, 100),</math> then their outer product <math>\mathbf{C}</math> is of order 5 with dimensions <math>(3, 5, 7, 10, 100).</math> If <math>\mathbf{A}</math> has a component {{math|1=''A''<sub>[2, 2, 4]</sub> = 11}} and <math>\mathbf{B}</math> has a component {{math|1=''B''<sub>[8, 88]</sub> = 13}}, then the component of <math>\mathbf{C}</math> formed by the outer product is {{math|1=''C''<sub>[2, 2, 4, 8, 88]</sub> = 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 <math>\mathbf{u} = \begin{bmatrix}1 & 2 & 3\end{bmatrix}^\textsf{T}</math> and <math>\mathbf{v} = \begin{bmatrix}4 & 5\end{bmatrix}^\textsf{T}</math>, we have:
:<math display="block">\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}</math>

In the case of column vectors, the Kronecker product can be viewed as a form of [[vectorization (mathematics)|vectorization]] (or flattening) of the outer product. In particular, for two column vectors <math>\mathbf{u}</math> and <math>\mathbf{v}</math>, we can write:
:<math display="block">\mathbf{u} \otimes_{\text{Kron}} \mathbf{v} = \operatorname{vec}(\mathbf{v} \otimes_\text{outer} \mathbf{u})</math>

(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
:<math display="block">\mathbf{u} \otimes_{\text{Kron}} \mathbf{v}^\textsf{T} = \mathbf u \mathbf{v}^\textsf{T} = \mathbf{u} \otimes_{\text{outer}} \mathbf{v}</math>

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 <math>\mathbf{A}</math> of size <math>m\times p</math> and <math>\mathbf{B}</math> of size <math>p\times n</math>, consider the [[matrix multiplication|matrix product]] <math>\mathbf{C} = \mathbf{A}\,\mathbf{B}</math> defined as usual as a matrix of size <math>m\times n</math>.

Now let <math>\mathbf a^\text{col}_k</math> be the <math>k</math>-th column vector of <math>\mathbf A</math> and let <math>\mathbf b^\text{row}_k</math> be the <math>k</math>-th row vector of <math>\mathbf B</math>. Then <math>\mathbf{C}</math> can be expressed as a sum of column-by-row outer products:
:<math display="block">\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 \mathbf b^\text{row}_k</math>
This expression has duality with the more common one as a matrix built with row-by-column [[inner product space|inner product]] entries (or [[dot product]]): <math>C_{ij} = \langle{\mathbf a^\text{row}_i,\,\mathbf b_j^\text{col}}\rangle</math>

This relation is relevant<ref>{{cite book |last2=Bau III |first2=David |last1=Trefethen |first1=Lloyd N. |author1-link=Lloyd N. Trefethen |title=Numerical linear algebra |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |isbn=978-0-89871-361-9 |year=1997}}</ref> in the application of the [[singular value decomposition#Separable models|Singular Value Decomposition (SVD)]] (and [[eigendecomposition of a matrix|Spectral Decomposition]] as a special case). In particular, the decomposition can be interpreted as the sum of outer products of each left (<math>\mathbf{u}_k</math>) and right (<math>\mathbf{v}_k</math>) singular vectors, scaled by the corresponding nonzero singular value <math>\sigma_k</math>:
:<math display="block">\mathbf{A} = \mathbf{U \Sigma V^T} = \sum_{k=1}^{\operatorname{rank}(A)}(\mathbf{u}_k \otimes \mathbf{v}_k) \, \sigma_k</math>

This result implies that <math>\mathbf{A}</math> can be expressed as a sum of rank-1 matrices with [[matrix norm#Norms induced by p-norms|spectral norm]] <math>\sigma_k</math> in decreasing order. This explains the fact why, in general, the last terms contribute less, which motivates the use of the [[singular value decomposition#Truncated SVD|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:
:<math display="block">\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}</math>

The outer product of tensors satisfies the additional [[associativity]] property:
:<math display="block">
  (\mathbf{u} \otimes \mathbf{v}) \otimes \mathbf{w} =
  \mathbf{u} \otimes (\mathbf{v} \otimes \mathbf{w})
</math>

===Rank of an outer product===
If '''u''' and '''v''' are both nonzero, then the outer product matrix '''uv'''<sup>T</sup> always has [[rank (linear algebra)|matrix rank]] 1. Indeed, the columns of the outer product are all proportional to '''u'''. 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 {{mvar|V}} and {{mvar|W}} be two [[vector space]]s. The outer product of <math>\mathbf v \in V</math> and <math>\mathbf w \in W</math> is the element <math>\mathbf v \otimes \mathbf w \in V \otimes W</math>.

If {{mvar|V}} is an [[inner product space]], then it is possible to define the outer product as a linear map {{math|''V'' → ''W''}}. In this case, the linear map <math>\mathbf x \mapsto \langle \mathbf v, \mathbf x\rangle</math> is an element of the [[dual space]] of {{mvar|V}}, as this maps linearly a vector into its underlying field, of which <math>\langle \mathbf v, \mathbf x\rangle</math> is an element. The outer product {{math|''V'' → ''W''}} is then given by
:<math display="block">(\mathbf w \otimes \mathbf v) (\mathbf x) = \left\langle \mathbf v, \mathbf x \right\rangle \mathbf w.</math>

This shows why a conjugate transpose of {{math|'''v'''}} is commonly taken in the complex case.

==In programming languages==
In some programming languages, given a two-argument function <code>f</code> (or a binary operator), the outer product, <code>f</code>, of two one-dimensional arrays, <code>A</code> and <code>B</code>, is a two-dimensional array <code>C</code> such that <code>C[i, j] = f(A[i], B[j])</code>. This is syntactically represented in various ways: in [[APL (programming language)|APL]], as the infix binary operator <syntaxhighlight lang=apl inline>∘.f</syntaxhighlight>; in [[J (programming language)|J]], as the postfix adverb <syntaxhighlight lang=j inline>f/</syntaxhighlight>; in [[R (programming language)|R]], as the function <syntaxhighlight lang=r inline>outer(A, B, f)</syntaxhighlight> or the special <syntaxhighlight lang=r inline>%o%</syntaxhighlight>;<ref>{{cite web |title=outer function {{!}} R Documentation |url=https://www.rdocumentation.org/packages/base/versions/3.6.2/topics/outer|access-date=2020-09-07 |website=rdocumentation.org}}</ref> in [[Wolfram Mathematica|Mathematica]], as <syntaxhighlight lang=mathematica inline>Outer[f, A, B]</syntaxhighlight>. In [[MATLAB]], the function <syntaxhighlight lang=matlab inline>kron(A, B)</syntaxhighlight> is used for this product. These often generalize to multi-dimensional arguments, and more than two arguments.

In the [[Python (programming language)|Python]] library [[NumPy]], the outer product can be computed with function <code>np.outer()</code>.<ref>{{cite web |title=numpy.outer — NumPy v1.19 Manual |url=https://numpy.org/doc/stable/reference/generated/numpy.outer.html |access-date=2020-09-07 |website=numpy.org}}</ref> In contrast, <code>np.kron</code> results in a flat array. The outer product of multidimensional arrays can be computed using <code>np.multiply.outer</code>.

==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]].<ref>{{cite book |last1=Steeb |first1=Willi-Hans |title=Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra |last2=Hardy |first2=Yorick |publisher=World Scientific |year=2011 |isbn=978-981-4335-31-7 |edition=2 |chapter=Applications (Chapter 3)}}</ref>

===Spinors===
Suppose {{math|''s'', ''t'', ''w'', ''z'' ∈ '''C'''}} so that {{math|(''s'', ''t'')}} and {{math|(''w'', ''z'')}} are in {{math|'''C'''<sup>2</sup>}}. Then the outer product of these complex 2-vectors is an element of {{math|M(2, '''C''')}}, the 2 × 2 complex matrices:
:<math display="block">\begin{pmatrix} sw & tw \\ sz & tz \end{pmatrix}.</math>
The [[determinant]] of this matrix is {{math|1=''swtz'' − ''sztw'' = 0}} because of the [[commutative property]] of {{math|'''C'''}}.

In the theory of [[spinors in three dimensions]], these matrices are associated with [[isotropic vector]]s due to this null property. [[Élie Cartan]] described this construction in 1937,<ref>[[Élie Cartan]] (1937) ''Lecons sur la theorie des spineurs'', translated 1966: ''The Theory of Spinors'', Hermann, Paris</ref> but it was introduced by [[Wolfgang Pauli]] in 1927<ref>Pertti Lounesto (1997) ''Clifford Algebras and Spinors'', page 51, [[Cambridge University Press]] {{ISBN|0-521-59916-4}}</ref> so that {{math|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 (logic)|and]]'' takes the place of multiplication. The outer product of two logical vectors {{math|(''u''<sub>i</sub>)}} and {{math|(''v''<sub>j</sub>)}} is given by the logical matrix <math>\left(a_{ij}\right) = \left(u_i \land v_j\right)</math>. This type of matrix is used in the study of [[binary relation]]s, and is called a ''rectangular relation'' or a '''cross-vector'''.<ref>[[Ki-Hang Kim]] (1982) ''Boolean Matrix Theory and Applications'', page 37, [[Marcel Dekker]] {{ISBN|0-8247-1788-0}}</ref>

==See also==
* [[Dyadics]]
* [[Householder transformation]]
* [[Norm (mathematics)]]
* [[Ricci calculus]]
* [[Scatter matrix]]

===Products===
* [[Cartesian product]]
* [[Cross product]]
* [[Exterior product]]
* [[Hadamard product (matrices)]]

===Duality===
* [[Bra–ket notation#Outer products|Bra–ket notation for outer product]]
* [[Complex conjugate]]
* [[Conjugate transpose]]
* [[Transpose]]

==References==
{{Reflist}}

==Further reading==
*{{cite book |last1=Carlen |first1=Eric |last2=Canceicao Carvalho |first2=Maria |title=Linear Algebra: From the Beginning |chapter=Outer Products and Orthogonal Projections |publisher=Macmillan |year=2006 |pages=217–218 |isbn=9780716748946 |chapter-url=https://books.google.com/books?id=EqGjBOXbPAoC&pg=PA217}}

{{Linear algebra}}

[[Category:Bilinear maps]]
[[Category:Higher-order functions]]
[[Category:Operations on vectors]]