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Unformatted text preview: Orthogonal Projections and Reflections (with exercises) by D. Klain Version 2010.01.23 Corrections and comments are welcome! Orthogonal Projections Let X 1 ,...,X k be a family of linearly independent (column) vectors in R n , and let W = Span( X 1 ,...,X k ) . In other words, the vectors X 1 ,...,X k form a basis for the k-dimensional subspace W of R n . Suppose we are given another vector Y ∈ R n . How can we project Y onto W orthogonally? In other words, can we find a vector ˆ Y ∈ W so that Y- ˆ Y is orthogonal (perpendicular) to all of W ? See Figure 1. To begin, translate this question into the language of matrices and dot products. We need to find a vector ˆ Y ∈ W such that ( Y- ˆ Y ) ⊥ Z, for all vectors Z ∈ W . (1) Actually, it’s enough to know that Y- ˆ Y is perpendicular to the vectors X 1 ,...,X k that span W . This would imply that (1) holds. (Why?) Expressing this using dot products, we need to find ˆ Y ∈ W so that X T i ( Y- ˆ Y ) = 0 , for all i = 1 , 2 ,...,k . (2) This condition involves taking k dot products, one for each X i . We can do them all at once by setting up a matrix A using the X i as the columns of A , that is, let A = X 1 X 2 ··· X k . Note that each vector X i ∈ R n has n coordinates, so that A is an n × k matrix. The set of conditions listed in (2) can now be re-written: A T ( Y- ˆ Y ) = 0 , which is equivalent to A T Y = A T ˆ Y . (3) 1 Figure 1: Projection of a vector onto a subspace. Meanwhile, we need the projected vector ˆ Y to be a vector in W , since we are projecting onto W . This means that ˆ Y lies in the span of the vectors X 1 ,...,X k . In other words, ˆ Y = c 1 X 1 + c 2 X 2 + ··· + c k X k = A c 1 c 2 . . . c k = AC. where C is a k-dimensional column vector. On combining this with the matrix equation (3) we have A T Y = A T AC. If we knew what C was then we would also know ˆ Y , since we were given the columns X i of A , and ˆ Y = AC . To solve for C just invert the k × k matrix A T A to get ( A T A )- 1 A T Y = C. (4) How do we know that ( A T A )- 1 exists? Let’s assume it does for now, and then address this question later on. Now finally we can find our projected vector ˆ Y . Since ˆ Y = AC , multiply both sides of (4) to obtain A ( A T A )- 1 A T Y = AC = ˆ Y . The matrix Q = A ( A T A )- 1 A T is called the projection matrix for the subspace W . According to our derivation above, the projection matrix Q maps a vector Y ∈ R n to its orthogonal projection (i.e. its shadow) QY = ˆ Y in the subspace W ....
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.
- Fall '08