outer product

Here, is the outer product of two vectors and .

Notice that the outer product operator is of course non-commutative.

The inner product is the trace of the outer product.

An important operation on arrays is the outer product.

In some contexts, this product is also referred to as outer product.

It is not the outer product of linear algebra.

It is common practice to extend the outer product on vectors to the entire algebra.

An alternative method is to express the matrix product in terms of the outer product.

The inner and outer products are associated with familiar concepts from standard vector algebra.

An initial Hessian estimate is constructed from the outer product of the gradient.

Please note that outer products apply to vectors.

The matrix can be written as the expected value of the outer product of with itself, namely:

Note that the outer product is defined for different dimensions, while the inner product requires the same dimension.

The map , representing scalar multiplication as a sum of outer products.

The result of applying the outer product to a pair of coordinate vectors is a matrix.

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

Rudelson and Vershynin give a result for matrices which are the outer product of two vectors.

Another generalization related to the outer product is the commutator product:

The term "inner product" is opposed to outer product, which is a slightly more general opposite.

The estimated standard errors were constructed from the matrix of outer products of the first partial derivatives.

The outer product is simply the Kronecker product, limited to vectors (instead of matrices).

A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:

For "outer product" in geometric algebra, see Exterior algebra.

For example, they do not support such useful and important operations as inner products, outer products, matrix multiplication, rotation, etc.

The outer product, tp' can then be subtracted from X leaving the residual matrix E.