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Unformatted text preview: Householder Reflectors The Householder reflector is arguably the most important tool in (dense) numerical linear algebra. Let u R n 1 . Then the Householder reflector defined by u is given by H = H ( u ) = I- uu t , where = 2 / ( u t u ) . Algebraically: H = H- 1 is a symmetric (Hermitian) rank-1 perturbation of I . Analytically: H is an orthogonal (unitary) matrix. Geometrically: Hv is the reflection of v about the hyperplane orthogonal to u (as a function: u H ( u ) has domain R P n- 1 , and as an operator: H : v Hv is an orthogonal reflector on R n ). Typically, H is used in matrix factorizations to introduce zeros into some other matrix. To see how it works, suppose we would like an arbitrary vector x to be sent to a multiple of some vector y under the action of H , i.e. find u such that Hx = y . Since H is orthogonal, k x k 2 = k Hx k 2 = | |k y k 2 , giving | | = k x k 2 / k y k 2 . If ( I- uu t ) x = y , then u = x- y , where = ( u t x )...
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- Fall '10