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The Householder transformation was introduced 1958 by Alston Scott Householder . It can be used to obtain a QR decomposition of a matrix.

1 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: Q² = I.

Furthermore, Q really reflects a point X (which we will identify with its position vector x) as describe above, since

Qx = x − 2 vv^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.

2 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) minorIn linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. Suppose A is an m ''n matrix and k is a positive integer not larger than m and n''. A k ''k minor of A is the determinant of a k ''k matrix obtained from A by deletings of that product. See the QR decomposition article for more.