Householder transformation

From Wikipedia, the free encyclopedia

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 in 1958 by Alston Scott Householder.

It can be used to obtain a QR decomposition as described in the QR algorithm of a matrix, bringing the matrix A to upper Hessenberg matrix form (which costs $\begin{matrix}\frac{2}{3}\end{matrix} n^3 + O(n^2)$ with a finite sequence of orthogonal similarity transforms.

Over general inner product spaces, this is known as the Householder operator.

[edit] Definition and properties

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

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

$Q = I - 2 vv^T.\,$

The Householder matrix has the following properties:

it is symmetric: $Q = Q^T\,$
it is orthogonal: $Q^{-1}=Q^T\,$
therefore it is also involutary: $Q^2=I\,$ .

Furthermore, $Q$ really reflects a point $X$ (which we will identify with its position vector $x$ ) as described above, since

$Qx = x-2vv^Tx = x - 2\langle v,x\rangle v,$

where $\langle \rangle$ denotes the dot product. Note that $\langle v, x\rangle$ is equal to the distance from X to the hyperplane.

[edit] Application

Main article: 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.