Finding Oblique Projector P for U, W Subspaces

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In summary, the task is to find the oblique projector P where range(P) is equal to range(U) and range(I-P) is equal to range(W). This can be achieved by using the Moore-Penrose inverse and adding the identity to fill up space in U. The solution can be found in "Generalized inverses: theory and applications" by AvAdi Ben-Israel, Thomas Nall Eden Greville.
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kalleC
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Homework Statement


Find the oblique projector P so where range(P) = range(U) and range(I-P) = range(W)


Homework Equations


P^2-P = 0
range(I-P) = null(P)


The Attempt at a Solution


It seems that U and W are complementary subspaces. According to:
http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.4500v4.pdf
U*(V*U)t*V
where t is the Moore-Penrose inverse. The task is to be done in Matlab. My problem is that V and U cannot be multiplied due to their sizes. The sizes are for example (5,3) and (5,8) with the rows in common. I tried adding on the identity to fill up space in U and also tried to add null space but this did not work. I always seem to end up with a projector P so that:
P^2-P = 0
PA = A
but PB != 0
and (I-P)B != B

Any ideas?
 
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p = [U 0] [U V]^-1

According to "Generalized inverses: theory and applications" by AvAdi Ben-Israel, Thomas Nall Eden Greville

0 is nullspace

And sure as hell it works =D
 

FAQ: Finding Oblique Projector P for U, W Subspaces

1. What is an oblique projector P for U, W subspaces?

An oblique projector P for U, W subspaces is a linear transformation that projects any vector onto the subspace U, while simultaneously making it orthogonal to the subspace W. It essentially splits a vector into two components, one that lies in U and one that is orthogonal to W.

2. Why is finding an oblique projector P for U, W subspaces important?

Finding an oblique projector P can be useful in a variety of applications, such as signal processing, image compression, and data analysis. It allows for a more efficient representation of data by reducing the dimensionality of a vector while preserving important information.

3. How is an oblique projector P for U, W subspaces calculated?

An oblique projector P for U, W subspaces can be calculated using the formula P = A(ATA)-1AT, where A is a matrix whose columns form an orthonormal basis for both U and W subspaces.

4. What is the difference between an oblique projector and an orthogonal projector?

An oblique projector P for U, W subspaces projects a vector onto the subspace U and makes it orthogonal to the subspace W. On the other hand, an orthogonal projector projects a vector onto a subspace while keeping it orthogonal to all other subspaces. In other words, an oblique projector is more specific than an orthogonal projector.

5. Can an oblique projector P for U, W subspaces exist for any pair of subspaces U and W?

No, an oblique projector P for U, W subspaces can only exist if U and W are complementary subspaces, meaning that their intersection is a zero vector. This condition ensures that the projection onto U does not affect the component that is orthogonal to W.

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