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HouseRefl - Householder Reectors The Householder reector is...

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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 - βuu t , where β = 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 = αy . Since H is orthogonal, x 2 = Hx 2 = | α | y 2 , giving | α | = x 2 / y 2 . If ( I - βuu t ) x = αy , then ηu = x - αy , where η = β ( u t x ) R . Since H ( u ) = H ( γu ), we may take u to be any (nonzero) multiple of x ± αy .
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