In linear algebra, the **outer product** of two coordinate vectors is a matrix. If the two 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

Given two vectors of size and respectively

their outer product, denoted is defined as the matrix obtained by multiplying each element of by each element of :

Or in index notation:

Denoting the dot product by if given an vector then If given a vector then

If and are vectors of the same dimension bigger than 1, then .

The outer product is equivalent to a matrix multiplication provided that is represented as a column vector and as a column vector (which makes a row vector).^{[2]}^{[3]} For instance, if and then^{[4]}

For complex vectors, it is often useful to take the conjugate transpose of denoted or :

If then one can take the matrix product the other way, yielding a scalar (or matrix):

which is the standard inner product for Euclidean vector spaces,

Multiplication of a vector by the matrix can be written in terms of the inner product, using the relation .

Given two tensors with dimensions and , their outer product is a tensor with dimensions and entries

For example, if is of order 3 with dimensions and is of order 2 with dimensions then their outer product is of order 5 with dimensions If has a component *A*_{[2, 2, 4]} = 11 and has a component *B*_{[8, 88]} = 13, then the component of formed by the outer product is *C*_{[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 of vectorization (or flattening) of the outer product. In particular, for two column vectors and , we can write:

Note that the order of the vectors is reversed in the right side of the equation.

Another similar identity that further highlights the similarity between the operations is

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 of size and of size , consider the matrix product defined as usual as a matrix of size .

Now let be the -th column vector of and let be the -th row vector of . Then can be expressed as a sum of column-by-row outer products:

Note the duality of this expression with the more common one as a matrix built with row-by-column inner product entries (or dot product):

This relation is relevant^{[6]} 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 () and right () singular vectors, scaled by the corresponding nonzero singular value :

This result implies that can be expressed as a sum of rank-1 matrices with spectral norm 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.

The outer product of vectors satisfies the following properties:

The outer product of tensors satisfies the additional associativity property:

If **u** and **v** are both nonzero, then the outer product matrix **uv**^{T} 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".)

Let V and W be two vector spaces. The outer product of and is the element .

If V is an inner product space, then it is possible to define the outer product as a linear map *V* → *W*. In which case, the linear map is an element of the dual space of V. The outer product *V* → *W* is then given by

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

In some programming languages, given a two-argument function `f`

(or a binary operator), the outer product of `f`

and 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`

.

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]}

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:

The determinant of this matrix is

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,^{[10]} but it was introduced by Wolfgang Pauli in 1927^{[11]} so that M(2, **C**) has come to be called Pauli 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 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 . This type of matrix is used in the study of binary relations, and is called a rectangular relation or a **cross-vector**.^{[12]}