In mathematics a '''Householder transformation''' in 3-dimensional space is the reflection (mathematics)|reflection of a Vector (spatial)|vector in a plane (mathematics)|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 $v$ (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 symmetrical: $Q\; =\; Q^T$
- 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 describe above since

- $Qx\; =\; x-2vv^Tx\; =\; x\; -\; 2v$

where < > denotes the

dot product Note that

$$ 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****) minor of that product See the QR decomposition article for more**