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

In summary, the conversation discusses the confusion around the projection matrix and its relation to the matrix A. The confusion arises due to the cancelation of inverse operations, which is only possible if matrix A is invertible. The conversation also discusses finding the matrix of projection for a given plane, which involves finding the projection of each basis vector onto the plane.
  • #1
EvLer
458
0
I am studying for exam and something does not make sense anymore:
looking at projection matrix, how come P=P2 where
P2 = A(ATA)-1ATA(ATA)-1AT
= A(ATA)-1AT
= 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 ATA 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...
 
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  • #2
Yes, although a projection map is not, in general, invertible, it's not too difficult to show that PTP always is.

Yes, (1, 1, 2) is normal to the given plane and, since that has length [itex]\sqrt{6}[/itex], [itex](1/\sqrt{6},1/\sqrt{6},2/\sqrt{6})[/itex] 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 [itex](1/\sqrt{6},1/\sqrt{6},2/\sqrt{6})[/itex] is [itex]1/\sqrt{6}[/itex], 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).
 
  • #3
thanks I think I got it now :)
 

What is a projection matrix?

A projection matrix is a square matrix that is used to transform a vector space into a subspace by projecting each vector onto a lower-dimensional subspace.

What are the applications of projection matrices?

Projection matrices are commonly used in computer graphics, computer vision, and machine learning to represent and manipulate three-dimensional objects.

How do you calculate a projection matrix?

To calculate a projection matrix, you first need to determine the basis vectors of the subspace onto which you want to project. Then, you can use the Gram-Schmidt process or the QR decomposition method to find the projection matrix.

What is the difference between an orthogonal and an oblique projection matrix?

An orthogonal projection matrix preserves the angles and lengths of the vectors, while an oblique projection matrix may distort them. In other words, an orthogonal projection is an isometry, while an oblique projection is not.

How is a projection matrix related to an identity matrix?

An identity matrix is a special case of a projection matrix, where the subspace onto which the vectors are projected is the same as the original vector space. In other words, an identity matrix represents a projection onto the entire vector space.

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