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Orthogonal Projection onto a Subspace

03/08/2011

Revisit:

  1. Let P be an orthogonal projection matrix, then P^T=P and P^2=P.
  2. x and y are two vectors. The projection of y onto x is \frac{xx^T}{x^Tx}y.

New question: Let A=(a_1,...,a_m)\in\mathbb{R}^{n\times m}, n\ge m. y is an vector. What is the projection of y onto the column space of A?

Answer: A(A^TA)^{-1}A^Ty

Proof: it is the same to find x to minimize \|Ax-y\|^2 which is a least squares problem. As we know, the answer is x=(A^TA)^{-1}A^Ty. Then the projection is Ax= A(A^TA)^{-1}A^Ty. Hence the projector is A(A^TA)^{-1}A^T.

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