Inlinear algebra, theouter product of twocoordinate 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 dimensionsn andm, then their outer product is ann ×m matrix. More generally, given twotensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as theirtensor product, and can be used to define thetensor algebra.

The outer product contrasts with:

• Thedot product (a special case of "inner product"), which takes a pair of coordinate vectors as input and produces ascalar
• TheKronecker product, which takes a pair of matrices as input and produces ablock matrix
• Standardmatrix multiplication

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

$${\displaystyle \mathbf{u}=\left[\begin{array}{c}{u}_1\\ u_2\\ ⋮\\ u_m\end{array}\right],\mathbf{v}=\left[\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\right]}$$

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]

Or, in index notation:

$${\displaystyle (\mathbf{u}\otimes \mathbf{v}{)}_{ij}=u_i{v}_j}$$

Denoting thedot 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${\displaystyle \mathbf{x}(\mathbf{u}\otimes \mathbf{v})=(\mathbf{x}\cdot \mathbf{u}){\mathbf{v}}^{\mathrm{T}}.}$

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 amatrix multiplication${\displaystyle \mathbf{u}{\mathbf{v}}^{\mathrm{T}},}$ 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${\displaystyle {\mathbf{v}}^{\mathrm{T}}}$ a row vector).^[2][3] For instance, if${\displaystyle m=4}$ and${\displaystyle n=3,}$ then^[4]

Forcomplex vectors, it is often useful to take theconjugate transpose of${\displaystyle \mathbf{v},}$ denoted${\displaystyle {\mathbf{v}}^{†}}$ or${\displaystyle {\left({\mathbf{v}}^{\mathsf{\text{T}}}\right)}^{\ast }}$:

$${\displaystyle \mathbf{u}\otimes \mathbf{v}=\mathbf{u}{\mathbf{v}}^{†}=\mathbf{u}{\left({\mathbf{v}}^{\mathsf{\text{T}}}\right)}^{\ast }.}$$

If${\displaystyle m=n,}$ then one can take the matrix product the other way, yielding a scalar (or${\displaystyle 1\times 1}$ matrix):

$${\displaystyle ⟨\mathbf{u},\mathbf{v}⟩={\mathbf{u}}^{\mathsf{\text{T}}}\mathbf{v}}$$

which is the standardinner product forEuclidean vector spaces,^[3] better known as thedot product. The dot product is thetrace of the outer product.^[5] Unlike the dot product, the outer product is not commutative.

Multiplication of a vector${\displaystyle \mathbf{w}}$ by the matrix${\displaystyle \mathbf{u}\otimes \mathbf{v}}$ can be written in terms of the inner product, using the relation${\displaystyle \left(\mathbf{u}\otimes \mathbf{v}\right)\mathbf{w}=\mathbf{u}⟨\mathbf{v},\mathbf{w}⟩}$.

Given two tensors${\displaystyle \mathbf{u},\mathbf{v}}$ with dimensions${\displaystyle (k_1,k_2,\dots ,k_m)}$ and${\displaystyle (l_1,l_2,\dots ,l_n)}$, their outer product${\displaystyle \mathbf{u}\otimes \mathbf{v}}$ is a tensor with dimensions${\displaystyle (k_1,k_2,\dots ,k_m,l_1,l_2,\dots ,l_n)}$ and entries

$${\displaystyle (\mathbf{u}\otimes \mathbf{v}{)}_{i_1,i_2,\dots i_m,j_1,j_2,\dots ,j_n}=u_{i_1,i_2,\dots ,i_m}{v}_{j_1,j_2,\dots ,j_n}}$$

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

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

If and, we have:

In the case of column vectors, the Kronecker product can be viewed as a form ofvectorization (or flattening) of the outer product. In particular, for two column vectors${\displaystyle \mathbf{u}}$ and${\displaystyle \mathbf{v}}$, we can write:

$${\displaystyle \mathbf{u}{\otimes }_{\text{Kron}}\mathbf{v}=\mathrm{vec}(\mathbf{v}{\otimes }_{\text{outer}}\mathbf{u})}$$

(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

$${\displaystyle \mathbf{u}{\otimes }_{\text{Kron}}{\mathbf{v}}^{\mathsf{\text{T}}}=\mathbf{u}{\mathbf{v}}^{\mathsf{\text{T}}}=\mathbf{u}{\otimes }_{\text{outer}}\mathbf{v}}$$

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

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

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

This expression has duality with the more common one as a matrix built with row-by-columninner product entries (ordot product):${\displaystyle C_{ij}=⟨{\mathbf{a}}_i^{\text{row}},{\mathbf{b}}_j^{\text{col}}⟩}$

This relation is relevant^[6] in the application of theSingular Value Decomposition (SVD) (andSpectral Decomposition as a special case). In particular, the decomposition can be interpreted as the sum of outer products of each left (${\displaystyle {\mathbf{u}}_k}$) and right (${\displaystyle {\mathbf{v}}_k}$) singular vectors, scaled by the corresponding nonzero singular value${\displaystyle {\sigma }_k}$:

$${\displaystyle \mathbf{A}=\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^{\mathbf{T}}=\sum _{k=1}^{\mathrm{rank}(A)}({\mathbf{u}}_k\otimes {\mathbf{v}}_k){\sigma }_k}$$

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

The outer product of vectors satisfies the following properties:

The outer product of tensors satisfies the additionalassociativity property:

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

Ifu andv are both nonzero, then the outer product matrixuv^T always hasmatrix rank 1. Indeed, the columns of the outer product are all proportional tou. Thus they are alllinearly 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".)

LetV andW be twovector spaces. The outer product of${\displaystyle \mathbf{v}\in V}$ and${\displaystyle \mathbf{w}\in W}$ is the element${\displaystyle \mathbf{v}\otimes \mathbf{w}\in V\otimes W}$.

IfV is aninner product space, then it is possible to define the outer product as a linear mapV →W. In this case, the linear map${\displaystyle \mathbf{x}\mapsto ⟨\mathbf{v},\mathbf{x}⟩}$ is an element of thedual space ofV, as this maps linearly a vector into its underlying field, of which${\displaystyle ⟨\mathbf{v},\mathbf{x}⟩}$ is an element. The outer productV →W is then given by

$${\displaystyle (\mathbf{w}\otimes \mathbf{v})(\mathbf{x})=⟨\mathbf{v},\mathbf{x}⟩\mathbf{w}.}$$

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

In some programming languages, given a two-argument functionf (or a binary operator), the outer product,f, of two one-dimensional arrays,A andB, is a two-dimensional arrayC such thatC[i, j] = f(A[i], B[j]). This is syntactically represented in various ways: inAPL, as the infix binary operator∘.f; inJ, as the postfix adverbf/; inR, as the functionouter(A,B,f) or the special%o%;^[7] inMathematica, asOuter[f,A,B]. InMATLAB, the functionkron(A,B) is used for this product. These often generalize to multi-dimensional arguments, and more than two arguments.

In thePython libraryNumPy, the outer product can be computed with functionnp.outer().^[8] In contrast,np.kron results in a flat array. The outer product of multidimensional arrays can be computed usingnp.multiply.outer.

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

Supposes,t,w,z ∈C so that(s,t) and(w,z) are inC². Then the outer product of these complex 2-vectors is an element ofM(2,C), the 2 × 2 complex matrices:

Thedeterminant of this matrix isswtz −sztw = 0 because of thecommutative property ofC.

In the theory ofspinors in three dimensions, these matrices are associated withisotropic vectors due to this null property.Élie Cartan described this construction in 1937,^[10] but it was introduced byWolfgang Pauli in 1927^[11] so thatM(2,C) has come to be calledPauli algebra.

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 alogical vector, a special case of alogical matrix. The logical operationand takes the place of multiplication. The outer product of two logical vectors(u_i) and(v_j) is given by the logical matrix${\displaystyle \left(a_{ij}\right)=\left(u_i\wedge v_j\right)}$. This type of matrix is used in the study ofbinary relations, and is called arectangular relation or across-vector.^[12]

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

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

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

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

• Bilinear maps
• Higher-order functions
• Operations on vectors

• Articles with short description
• Short description is different from Wikidata