A rank one matrix can be written in a special form Any rank one matrix can be written in the form $uv^{t}$, where $u,v$ are column vectors considered as matrices and $t$ denotes transposition. 
Why? How?

One way to see this is with SVD. This is overkill, though.
Another approach: suppose that $A$ is a rank $1$ matrix. Then $A$ has at least one non-zero row. Call this row $v^T$. Every row of $A$ must be a multiple of this row. That is, there exist coefficients $u_i$ such that the rows of $A$ are exactly $u_1v^T,u_2v^T,\dots u_nv^T$. 
This is exactly the same as saying that $A= uv^T$, where $u$ is the (column-)vector $(u_1,\dots,u_n)$.
