Projections

# Projections - Projections With the inner product we...

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Projections With the inner product <x, y> , we have angles ( <x, y> = x 2 y 2 cos ( θ )), and can speak of orthogonality: x y ⇐⇒ <x, y> = 0. Here we will consider the standard inner product for R n : <x, y> x t y , but more general inner products can be very useful in many applications and algorithm development. If S is a subspace of R n (write S R n ), we say x S if x is orthogonal to every element of S . The subspace S R n is called the orthogonal complement of S S ≡ { x R n : x S } , and R n = S S is a direct sum decomposition of R n into complementary subspaces in such a way that each x R n has the unique factorization x = u + v , with u S and v S . In this setting, u is the orthogonal projection of x onto S . In this note, we will be looking at a transformation P , which satisfies x R n , Px = u, the orthogonal projection of x onto S. If x S , then we should have Px = x (right?), so P should satisfy P 2 = P , with Range( P ) = S . The requirement that Px = u v , with u S and v S , combined with x t y = y t x forces P to be self-adjoint (in matrix language over R , P t = P ). Any linear transformation P which satisfies 1. Range( P ) = S , 2. P 2 = P, and 3. P t = P is called an
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