# Householder transformation

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 [itex]v[itex] (a vector with length 1), that is orthogonal to the hyperplane.

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

[itex]Q = I - 2 vv^T[itex].

The Householder matrix has the following properties:

• it is symmetrical: [itex]Q = Q^T[itex]
• it is orthogonal: [itex]Q^{-1}=Q^T[itex]
• therefore it is also involutary: [itex]Q^2=I[itex].

Furthermore, [itex]Q[itex] really reflects a point X (which we will identify with its position vector [itex]x[itex]) as describe above, since

[itex]Qx = x-2vv^Tx = x - 2v[itex],

where < > denotes the dot product. Note that [itex][itex] 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

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy