# Rank of $vv^\top + ww^\top$

Given two non-colinear real unit vectors $v,w$, I believe the rank of $M=vv^\top + ww^\top$ is 2 and I'd like to prove it. $vv^\top$ and $ww^\top$ are obviously of rank one, $v$ is not in the kernel of $M$ because $vv^\top v=v$ and $\|ww^\top v\|<1$ (because $v$ and $w$ are not colinear), same with $w$. So the rank of $M$ is at least 2 [edit just realised this argument is wrong; the non-colinearity of the images is required]. Also, by subadditivity of the rank, $\text{rank}(vv^\top + ww^\top )\leq \text{rank}(vv^\top) +\text{rank}(ww^\top )$. Do you see any mistakes/do you have another proof?

Choose a basis in which $e_1=v$ and $e_2=w$. Then $vv^\top+ww^\top=E_{11}+E_{22}$, and it has rank two.
Hint: Take a vector $x$ and write down an expression for $Mx$. Then, regardless of $x$, $Mx$ is a linear combination of...
• ... its projection on $v$ and its projection on $w$. Sep 8 '16 at 19:37
• So every value of $Mx$ is a linear combination of $v$ and $w$, and hence its rank is 2. Sep 8 '16 at 19:43
Yet another proof: note that $$vv^T + ww^T = \pmatrix{v&w} \pmatrix{v & w}^T$$ and for any matrix $M$, $M$ has the same rank as $MM^T$.