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1.12 Projections

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Definition 1

A linear transformation P : X → X is called a projection if P ( x ) = x ∀ x ∈ R ( P ) , i.e, P ( P ( x ) ) = P ( x ) ∀ x ∈ X .

P : R 3 → R 3 , P ( x 1 , x 2 , x 3 ) = ( x 1 , x 2 , 0 )

An illustration showing the projection operator that maps x to the x_1 x_2 - plane.

Definition 2

If P is a projection operator on an inner product space V , we say that P is an orthogonal projection if R ( P ) ⊥ N ( P ) , i.e., ⟨ x , y ⟩ = 0 ∀ x ∈ R ( P ) , y ∈ N ( P ) .

If P is an orthogonal projection, then for any x ∈ V we can write:

x = P x + ( I - P ) x

where P x ∈ R ( P ) and ( I - P ) x ∈ N ( P ) (since P ( I - P ) x = P x - P ( P x ) = P x - P x = 0 .)

Now we see that the solution to our “best approximation in a linear subspace” problem is an orthogonal projection: we wish to find a P such that R ( P ) = A .

Illustrations of the projection operator for a 1-D subspace of R2 and a 2-D subspace of R3. The best approximation to a point from the subspace is given by a projection of the point into that subspace.

The question is now, how can we design such a projection operator?
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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