• 107328  Infos

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^{-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 (ii) minor of that product See the QR decomposition article for more