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 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 - 2vv^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: Q² = 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 - 2 < v,x > v,

where < > denotes the dot product. Note that < v,x > 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.