The outer product of column vectors 
and 
over a field 
is the 
matrix
![uv^T=[u_1v_1 ... u_1v_n; | ... |; u_mv_1 ... u_mv_n].](/images/equations/OuterProduct/NumberedEquation1.svg) |
Thus its 
th
entry is 
. Over the complex
numbers, some authors use the conjugate transpose 
instead, especially when the outer
product is associated with an inner product.
An outer-product matrix has matrix rank at most 1, and has matrix rank 1 when both
vectors are nonzero. The outer product is the matrix representation
of the tensor product 
after bases have been chosen.
