Outer product

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
Definition.
Given two vectors
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:
If formula_4 and formula_5 are vectors of the same dimension, then formula_6.
The outer product is equivalent to a matrix multiplication uvT, 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
For complex vectors, it is often useful to take the conjugate transpose of v, denoted formula_8or formula_9:
Contrast with Euclidean inner product.
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, better known as the dot product. The inner product is the trace of the outer product. Unlike the inner product, the outer product is not commutative.
Multiplication of a vector formula_12 by the matrix formula_13 can be written in terms of the inner product, using the relation formula_14.
The outer product of tensors.
Given two tensors u, v with dimensions formula_15 and formula_16, their outer product formula_13 is a tensor with dimensions formula_18 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 formula_20 and formula_21, 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 formula_4 and formula_5, 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.
Properties.
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.
Definition (abstract).
Let "V" and "W" be two vector spaces. The outer product of formula_29 and formula_30 is the element formula_31.
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 formula_32 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 codice_1 (or a binary operator), the outer product of codice_1 and two one-dimensional arrays codice_3 and codice_4 is a two-dimensional array codice_5 such that codice_6. 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 codice_7.
In contrast, codice_8 results in a flat array.
The outer product of multidimensional arrays can be computed using codice_9.
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 ℂ2. Then the outer product of these complex 2-vectors is an element of M(2, ℂ), the 2 × 2 complex matrices:
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 formula_35. This type of matrix is used in the study of binary relations, and is called a rectangular relation or a cross-vector.