How does the projection matrix work and when can cancellations be made?

[EvLer]

I am studying for exam and something does not make sense anymore:
looking at projection matrix, how come P=P² where
P² = A(A^TA)^-1A^TA(A^TA)^-1A^T
= A(A^TA)^-1A^T
= P

but then they also say that cancelations (like distributing inverse operation and having AA^-1 = I type things) are possible only if A is invertible, so does that mean that A^TA is invertible?
Could someone "unconfuse" me please?

EDIT: might as well ask this:
find matrix of projection p on plane x+y+2z = 0
my attempt: I know that the normal vector is (1, 1, 2) and then... not sure where to go with that...

 

[HallsofIvy]

Yes, although a projection map is not, in general, invertible, it's not too difficult to show that P^TP always is.

Yes, (1, 1, 2) is normal to the given plane and, since that has length $\sqrt{6}$, $(1/\sqrt{6},1/\sqrt{6},2/\sqrt{6})$ is a unit vector normal to the plane.

The standard way to find the matrix for a linear transformation (in a given basis) is to see what it does to each of the basis vectors. Since the dot product of (1, 0, 0) with $(1/\sqrt{6},1/\sqrt{6},2/\sqrt{6})$ is $1/\sqrt{6}$, its projection onto that vector is (1/6, 1/6, 1/3) and so its projection onto the plane is (1, 0, 0)- (1/6, 1/6, 1/3)= (5/6, -1/6, -1/3). Similarly, the projection of (0, 1, 0) onto the plane is (-1/6, 5/6, -1/3) and the projection of (0, 0, 1) onto the plane is (-1/3, -1/3, 2/3). The matrix representation of the projection is the matrix having those vectors as columns (in that order).

 

[EvLer]

thanks I think I got it now :)