Outer product on arvau

Wikipedia / Outer product

In linear algebra, the outer product of two coordinate 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 dimensions m and n, then their outer product is an m × n 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
The Hadamard product, which is the element-wise product
Standard matrix multiplication

Definition

Given two column vectors, and of size their outer product sometimes denoted is the matrix product of the column vector times the row vector matrix whose entries are each possible product of one element of times one element of :

The outer product of two vectors always has rank , and conversely every matrix of rank can be written as an outer product.

For complex-valued vectors, it is common to define the outer product in terms of the conjugate transpose of denoted The conjugate transpose of a matrix results from taking the complex conjugate of each entry in the transpose of a matrix.

Contrast with Euclidean inner product

This is the standard inner product (or dot product) for the Euclidean vector space . The dot product is the trace of the outer product. Unlike the dot product, the outer product is not commutative.

The outer product of tensors

has a component has a component 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.

In the case of column vectors, the Kronecker product can be viewed as a form of vectorization

(The order of the vectors is reversed on the right side of the equation.)

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

, consider the matrix product.

This expression has duality with the more common one as a matrix built with row-by-column inner product entries (or dot product

This relation is relevant in the application of the Singular Value Decomposition (SVD) (and Spectral Decomposition

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.

Properties

The outer product of tensors satisfies the additional associativity

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 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 and be two vector spaces.

If is an inner product space, then it is possible to define the outer product as a linear map is an element of the dual space of

This shows why a conjugate transpose of 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, f, of 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 ; in J, as the postfix adverb ; in R, as the function or the special ; in Mathematica, as . In MATLAB, the function 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.

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 so that and are in . Then the outer product of these complex 2-vectors is an element of The determinant of this matrix is because of the commutative property of .

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 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 and . This type of matrix is used in the study of binary relations, and is called a rectangular relation or a cross-vector.