1. ## Rank-1 Projections

Show that if u has a unit length, then the rank-1 matrix P=uu^t is a projection matrix, meaning P^2=P and P^t=P. By choosing u = a / ||a||, P becomes the projection onto the line through a, and Pb is the point p = xa. Rank-1 projections correspond exactly to a least squares problem in 1 unknown.

OK... I don't really know what they're talking about... The only relevant equation I have is P=A(A^tA)^-1A^t. I don't know how to apply it. Please help!

2. Anyone? I know this question is not that difficult, but I think I'm missing something...

3. Originally Posted by veronicak5678
Show that if u has a unit length, then the rank-1 matrix P=uu^t is a projection matrix, meaning P^2=P and P^t=P. By choosing u = a / ||a||, P becomes the projection onto the line through a, and Pb is the point p = xa. Rank-1 projections correspond exactly to a least squares problem in 1 unknown.

OK... I don't really know what they're talking about... The only relevant equation I have is P=A(A^tA)^-1A^t. I don't know how to apply it. Please help!
I'm not entirely sure what's going on either but if $\displaystyle P=uu^{T}$ then $\displaystyle P^2=(uu^{T})(uu^{T})=u(u^{T}u)u^{T}=uu^{T}$ since $\displaystyle u^{T}u=u\cdot u=\|u\|^2=1$

4. That does help. Thanks for answering!