Outer product

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In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix. The name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar.

Other generalizations of the outer product are possible to more general vector objects such as tensors. The outer product is also a higher-order function in some computer programming languages such as APL and Mathematica.

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[edit] Definition (matrix multiplication)

The outer product of vectors is a special case of the Kronecker product of matrices.

Given the m\times 1 column vector \mathbf{u} and the 1 \times n row vector \mathbf{v}, their outer product \mathbf{u} \otimes \mathbf{v} is defined as the m\times n matrix \mathbf{A} resulting from

\mathbf{u} \otimes \mathbf{v} = \mathbf{A} = \mathbf{u} \mathbf{v}

where the tensor product is just multiplication of vectors.

In coordinates:

\begin{bmatrix}b_1 \\ b_2 \\ b_3 \\ b_4\end{bmatrix} \otimes \begin{bmatrix}a_1 & a_2 & a_3\end{bmatrix} = \begin{bmatrix}a_1b_1 & a_2b_1 & a_3b_1 \\ a_1b_2 & a_2b_2 & a_3b_2 \\ a_1b_3 & a_2b_3 & a_3b_3 \\ a_1b_4 & a_2b_4 & a_3b_4\end{bmatrix}


For complex vectors, it is customary to use the complex conjugate of \mathbf{v} (denoted \bar \mathbf{v}), as one thinks of row vectors as elements of the complex conjugate vector space of the dual vector space:

\mathbf{u} \otimes \mathbf{v} = \mathbf{A} = \mathbf{u} \bar \mathbf{v}

If \mathbf{v} is instead a column vector, the definition becomes:

\mathbf{u} \otimes \mathbf{v} = \mathbf{A} = \mathbf{u} \mathbf{v}^*

where \mathbf{v}^* is the conjugate transpose of \mathbf{v}.

[edit] Contrast with inner product

If \mathbf{v} is a row vector, and m = n, then one can take the product the other way, yielding a scalar (or 1 \times 1 matrix):

\left\langle \mathbf{u}, \mathbf{v}\right\rangle = \left\langle \mathbf{v}, \mathbf{u}\right\rangle = \mathbf{v} \mathbf{u}

which is the standard inner product for Euclidean vector spaces, better known as the dot product.

[edit] Definition (abstract)

Given a vector v \in V and a covector w^* \in W^*, the tensor product v \otimes w^* gives a map A\colon W \to V, under the isomorphism \mathrm{Hom}(W,V) = W^* \otimes V.

Concretely, given w \in W,

A(w): = w * (w)v

where w * (w) is w * evaluated on w, which yields a scalar, which then multiplies v.

Alternately, it's the composition of w^*\colon W \to K with v\colon K \to V.

If W = V, then one can also pair the covector w*∈V* with the vector v∈V via (w^*,v)\mapsto w^*(v), which is the duality pairing between V and its dual, sometimes called the inner product.

[edit] Definition (tensor multiplication)

The outer product on tensors is typically referred to as the tensor product. 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. .

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

  1. The vector v can be interpreted 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).
  2. 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.
  3. It is possible to 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).

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 computing physical quantities (e.g. the tensor of inertia), and 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.

[edit] See also

[edit] Products

[edit] Duality

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