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Encyclopedia > Outer product

Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. The cardinality of these operations is that of cartesian products. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X Ã— Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after RenÃ© Descartes...

Outer product is also a higher-order function in computer programming languages such as APL. Here, the cardinality of the results produced by this operation is that of cartesian products. In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... APL (for A Programming Language) is an array programming language based on a notation invented in 1957 by Kenneth E. Iverson while at Harvard University. ...

## Contents

Given the M-by-1 column vector v and the N-by-1 column vector u, the outer product is defined as the M-by-N matrix A resulting from In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In linear algebra, a column vector is an m Ã— 1 matrix, i. ...

$bold{A} = bold{v} otimes bold{u} = bold{v} bold{u^{*T}} = bold{v} bold{u^{H}}$

where $bold{u^{*T}}$ or equivalently $bold{u^{H}}$ indicates the conjugate transpose operator applied to vector u. In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

## Definition (Tensor operation)

Given a tensor a with rank q and dimensions (i 1, ..., i q), and a tensor b with rank r and dimensions (j 1, ..., j r), their outer product c has rank q+r and dimensions (k 1, ..., k q+r) which are the i  dimensions followed by the j  dimensions. For example, if A has rank 3 and dimensions (357) and B has rank 2 and dimensions (10100), their outer product c has rank 5 and dimensions (35710100). If A[224] = 11 and B[888]= 13 then C[224888] = 143. In other words, outer product on tensors is simply the tensor product. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Note: This is a fairly abstract mathematical approach to tensors. ... 2-dimensional renderings (ie. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

To understand the matrix definition of outer product in terms of the tensor definition of outer product:

1. You can interpret the vector v as a rank 1 tensor with dimension (M), and the vector u as a rank 1 tensor with dimension (N). The result is a rank 2 tensor with dimension (MN).
2. The result of an inner product between two tensors of rank q and r is the greater of q+r-2 and 0. Thus, the inner product of two matrices has the same rank as the outer product (or tensor product) of two vectors.
3. You can add arbitrarily many leading or trailing 1 dimensions to a tensor without fundamentally altering its structure. These 1 dimensions would alter the character of operations on these tensors, so any resulting equivalences should be expressed explicitly.
4. The inner product of two matrices V with dimensions (d, e) and U with dimensions (e, f) is $sum_{j = 1}^e V_i,_j U_j,_k$ where $i in {1..d}$ and $k in {1..f}$, For the case where e =1, the summation is trivial (involving only a single term). QED

It should be emphasized that the term "rank" is being used in its tensor sense, and should not be interpreted as matrix rank. // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

## Applications

The outer product is useful in performing transform operations in digital signal processing and digital image processing. It is also useful in statistical analysis for computing the covariance and auto-covariance matrices for two random variables. Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... It has been suggested that this article or section be merged into image processing. ... Statistics is the science and practice of developing knowledge through the use of empirical data expressed in quantitative form. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

Results from FactBites:

 Outer product - Wikipedia, the free encyclopedia (433 words) Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. Outer product is also a higher-order function in computer programming languages such as APL. The outer product is useful in performing transform operations in digital signal processing and digital image processing.
 Cross product - Wikipedia, the free encyclopedia (1611 words) In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. There is however the wedge product, which has similar properties, except that the wedge product of two vectors is now a 2-vector instead of an ordinary vector. In the context of multilinear algebra, it is possible to define a generalized cross product in terms of parity such that the generalized cross product between two vectors of dimension n is a tensor of rank n−2.
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