Eikonal Blog

2010.01.30

Householder transformation

Filed under: mathematics — Tags: , — sandokan65 @ 18:08

Definition: For an Euclidean vector v, the Householder matrix is defined as:

    P :\equiv {\bf 1} - 2 \frac{\underline{v}\otimes\underline{v}}{||\underline{v}||^2}

It expresses the mirroring transformation:

  • P \underline{v} = -\underline{v},
  • P \underline{u} = +\underline{u}; (\forall \underline{u} \perp \underline{v}).
  • P \underline{x} = - \underline{v}\frac{\underline{v}\cdot\underline{x}}{||\underline{v}||^2} + \left(1-\frac{\underline{v}\otimes\underline{v}}{||\underline{v}||^2}\right)\underline{x} = - \underline{x}_{||v} + \underline{x}_{\perp v} for an arbitrary vector \underline{x}.

Properties: P^T=P, P^2 = {\bf 1}.

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