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Householder TransformationIn 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 (a vector with length 1), that is orthogonal to the hyperplane. If is given as a column unit vector and is the identity matrix the linear transformation described above is given by the Householder matrix ( denotes the transpose of the vector ) - .
The Householder matrix has the following properties: - it is symmetrical:
- it is orthogonal:
- therefore it is also involutary: .
Furthermore, really reflects a point X (which we will identify with its position vector ) as describe above, since - ,
where < > denotes the dot product. Note that is equal to the distance of X to the hyperplane. 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.
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