HouseRefl - Householder Reflectors The Householder...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

Ask a homework question - tutors are online