Outer product

Summary

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 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 edit

Given two vectors of size   and   respectively

 

their outer product, denoted   is defined as the   matrix   obtained by multiplying each element of   by each element of  :[1]

 

Or, in index notation:

 

Denoting the dot product by   if given an   vector   then   If given a   vector   then  

If   and   are vectors of the same dimension bigger than 1, then  .

The outer product   is equivalent to a matrix multiplication   provided that   is represented as a   column vector and   as a   column vector (which makes   a row vector).[2][3] For instance, if   and   then[4]

 

For complex vectors, it is often useful to take the conjugate transpose of   denoted   or  :

 

Contrast with Euclidean inner product edit

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,[3] better known as the dot product. The dot product is the trace of the outer product.[5] Unlike the dot product, the outer product is not commutative.

Multiplication of a vector   by the matrix   can be written in terms of the inner product, using the relation  .

The outer product of tensors edit

Given two tensors   with dimensions   and  , their outer product   is a tensor with dimensions   and entries

 

For example, if   is of order 3 with dimensions   and   is of order 2 with dimensions   then their outer product   is of order 5 with dimensions   If   has a component A[2, 2, 4] = 11 and   has a component B[8, 88] = 13, then the component of   formed by the outer product is C[2, 2, 4, 8, 88] = 143.

Connection with the Kronecker product edit

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:

 

(The order of the vectors is reversed on 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.

Connection with the matrix product edit

Given a pair of matrices   of size   and   of size  , consider the matrix product   defined as usual as a matrix of size  .

Now let   be the  -th column vector of   and let   be the  -th row vector of  . Then   can be expressed as a sum of column-by-row outer products:

 

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[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 ( ) and right ( ) singular vectors, scaled by the corresponding nonzero singular value  :

 

This result implies that   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 edit

The outer product of vectors satisfies the following properties:

 

The outer product of tensors satisfies the additional associativity property:

 

Rank of an outer product edit

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".)

Definition (abstract) edit

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 VW. 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   is an element. The outer product VW is then given by

 

This shows why a conjugate transpose of v is commonly taken in the complex case.

In programming languages edit

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 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().[8] In contrast, np.kron results in a flat array. The outer product of multidimensional arrays can be computed using np.multiply.outer.

Applications edit

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.[9]

Spinors edit

Suppose s, t, w, zC 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:

 

The determinant of this matrix is swtzsztw = 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. Élie Cartan described this construction in 1937,[10] but it was introduced by Wolfgang Pauli in 1927[11] so that M(2,C) has come to be called Pauli algebra.

Concepts edit

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.[12]

See also edit

Products edit

Duality edit

References edit

  1. ^ Lerner, R. G.; Trigg, G. L. (1991). Encyclopaedia of Physics (2nd ed.). VHC. ISBN 0-89573-752-3.
  2. ^ Lipschutz, S.; Lipson, M. (2009). Linear Algebra. Schaum’s Outlines (4th ed.). McGraw-Hill. ISBN 978-0-07-154352-1.
  3. ^ a b Keller, Frank (February 23, 2020). "Algebraic Properties of Matrices; Transpose; Inner and Outer Product" (PDF). inf.ed.ac.uk. Archived (PDF) from the original on 2017-12-15. Retrieved September 6, 2020.
  4. ^ James M. Ortega (1987) Matrix Theory: A Second Course, page 7, Plenum Press ISBN 0-306-42433-9
  5. ^ Stengel, Robert F. (1994). Optimal Control and Estimation. New York: Dover Publications. p. 26. ISBN 0-486-68200-5.
  6. ^ Trefethen, Lloyd N.; Bau III, David (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-361-9.
  7. ^ "outer function | R Documentation". rdocumentation.org. Retrieved 2020-09-07.
  8. ^ "numpy.outer — NumPy v1.19 Manual". numpy.org. Retrieved 2020-09-07.
  9. ^ 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. ISBN 978-981-4335-31-7.
  10. ^ Élie Cartan (1937) Lecons sur la theorie des spineurs, translated 1966: The Theory of Spinors, Hermann, Paris
  11. ^ Pertti Lounesto (1997) Clifford Algebras and Spinors, page 51, Cambridge University Press ISBN 0-521-59916-4
  12. ^ Ki Hang Kim (1982) Boolean Matrix Theory and Applications, page 37, Marcel Dekker ISBN 0-8247-1788-0

Further reading edit

  • Carlen, Eric; Canceicao Carvalho, Maria (2006). "Outer Products and Orthogonal Projections". Linear Algebra: From the Beginning. Macmillan. pp. 217–218. ISBN 9780716748946.