Vector outer product is denoted by 
or 
. Vector outer product is also called vector matrix product because the result of the vector multiplication is a matrix. It can only be performed for two vectors of any size, not necessarily the same size.
Algebraically, the vector outer product is a multiplication of a column vector by a row vector 
to obtain a matrix provided by the formula below
Example
Suppose we have 
and 
, the vector outer products are
Properties
Some important properties of vector outer product are
- Vector outer product is a not commutative operation. If you reverse the order you will get different result and even different size
. - If matrix is the results of a vector product
then multiplication of that matrix with the second factor produces a vector of the first factor scaled by the square norm of the second factor 
.
Originally from http://people.revoledu.com/kardi/tutorial/LinearAlgebra/VectorOuterProduct.html
liuzhiguang 