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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
- Q = I - 2vvT.
The Householder matrix has the following properties:
- it is symmetrical: Q = QT
- it is orthogonal: Q - 1 = QT
- 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 - 2vvTx = 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.
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