If the two coordinate vectors have dimensions n and m, then their outer product is an n × m matrix.
The outer product contrasts with: Given two vectors of size
:[1] Or, in index notation: Denoting the dot product by
then[4] For complex vectors, it is often useful to take the conjugate transpose of
then one can take the matrix product the other way, yielding a scalar (or
can be written in terms of the inner product, using the relation
The outer product and Kronecker product are closely related; in fact the same symbol is commonly used to denote both operations.
, 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.
, we can write: (The order of the vectors is reversed on the right side of the equation.)
The middle expression uses matrix multiplication, where the vectors are considered as column/row matrices.
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 (
can be expressed as a sum of rank-1 matrices with spectral norm
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 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.
If V is an inner product space, then it is possible to define the outer product as a linear map V → W. In this case, the linear map
is an element of the dual space of V, as this maps linearly a vector into its underlying field, of which
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, 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 ∘.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 the Python library NumPy, the outer product can be computed with function np.outer().
The outer product of multidimensional arrays can be computed using np.multiply.outer.
These applications are found in quantum theory, signal processing, and image compression.
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 swtz − sztw = 0 because of the commutative property of C. In the theory of spinors in three dimensions, these matrices are associated with isotropic vectors due to this null property.
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.