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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, 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
their outer product, denoted , is defined as the matrix A obtained by multiplying each element of u by each element of v:
Or in index notation:
The outer product is equivalent to a matrix multiplicationuvT, provided that u is represented as a column vector and v as a column vector (which makes vT a row vector). For instance, if and , then
Given two tensors u, v with dimensions and , their outer product is a tensor with dimensions and entries
For example, if A is of order 3 with dimensions and B is of order 2 with dimensions , then their outer product C is of order 5 with dimensions . If A has a component and B has a component , then the component of C formed by the outer product is .
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 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.
The outer product of vectors satisfies the following properties:
The outer product of tensors satisfies the additional associativity property:
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".)
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 programming languages
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%; 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.
In contrast, np.kron results in a flat array.
The outer product of multidimensional arrays can be computed using np.multiply.outer.
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 (ui) and (vj) 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.
^Steeb, Willi-Hans; Hardy, Yorick (2011). "Applications (Chapter 3)". Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra (2 ed.). World Scientific. ISBN981-4335-31-2.
^Élie Cartan (1937) Lecons sur la theorie des spineurs, translated 1966: The Theory of Spinors, Hermann, Paris