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# Householder transformation

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: $Q^\left\{-1\right\}=Q^T$
• 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 (ii) minor of that product See the QR decomposition article for more