# Rank of the Outer Product of two Vectors

I have come across the statement that the rank of the outerproduct of two vectors is always one - but why is that?

Thanks

• This depends on the context. And what is the outer product, the wedge product $a\wedge b$ or the dyadic product $a\cdot b^\top$? Feb 25, 2014 at 10:49
• It would be the dyadic product. The context was just a remark on the slides we received.
– user66280
Feb 26, 2014 at 8:20

Outer Product generates the matrix whose first row is $u_1(v_1,v_2,..,v_n)$ and the ith row is $u_i(v_1,v_2,..,v_n)$. So the rows are the vector $(v_1,v_2,..,v_n)$ multiplied by scalars. So this itself is the basis.Hence dimension is 1.
To put it into a more philosophical frame: The rank $r$ of a matrix $A$ is the minimal number of dyadic products $u_kv_k^\top$ required to express the matrix as
$$A=\sum_{k=1}^r u_kv_k^\top.$$
Obviously, such representations exist as $A=\sum_{k=1}^n a_ke_k^\top$ where $a_k$ are the columns of $A$ and $e_k$ the canonical basis vectors of length $n$.
The outer product in its usual meaning is the anti-symmetric tensor product or wedge product $u\wedge v=\frac12(u\otimes v-v\otimes u)$. This obviously is either zero if $u\sim v$ or has rank $2$.