Householder transformation

From Academic Kids

In mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane. In general Euclidean space it is a linear transformation that describes a reflection in a hyperplane (containing the origin).

The Householder transformation was introduced 1958 by Alston Scott Householder. It can be used to obtain a QR decomposition of a matrix.

Definition and properties

The reflection hyperplane can be defined by a unit vector <math>v<math> (a vector with length 1), that is orthogonal to the hyperplane.

If <math>v<math> is given as a column unit vector and <math>I<math> is the identity matrix the linear transformation described above is given by the Householder matrix (<math>v^T<math> denotes the transpose of the vector <math>v<math>)

<math>Q = I - 2 vv^T<math>.

The Householder matrix has the following properties:

  • it is symmetrical: <math>Q = Q^T<math>
  • it is orthogonal: <math>Q^{-1}=Q^T<math>
  • therefore it is also involutary: <math>Q^2=I<math>.

Furthermore, <math>Q<math> really reflects a point X (which we will identify with its position vector <math>x<math>) as describe above, since

<math>Qx = x-2vv^Tx = x - 2v<math>,

where < > denotes the dot product. Note that <math><math> is equal to the distance of X to the hyperplane.

Application: QR decomposition

Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the (i,i) minors of that product. See the QR decomposition article for more.de:Householdertransformation it:Trasformazione di Householder

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