Outer product

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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.

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.

1 Definition (Matrix operation)
2 Definition (Tensor operation)
3 Applications
4 See also

[edit] Definition (Matrix operation)

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

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

where $\bold{u^{*T}} \ \$ or equivalently $\ \ \bold{u^{H}}$ indicates the conjugate transpose operator applied to vector u.

[edit] 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 (3, 5, 7) and B has rank 2 and dimensions (10, 100), their outer product c has rank 5 and dimensions (3, 5, 7, 10, 100). If A_[2, 2, 4] = 11 and B_[8, 88]= 13 then C_{[2, 2, 4, 8, 88]} = 143. In other words, outer product on tensors is simply the tensor product.

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

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 (M, N).
The rank of 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.
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.
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.

[edit] 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.