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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 in 1958 by Alston Scott Householder.
It can be used to obtain a QR decomposition as described in the QR algorithm of a matrix, and to bring a matrix A to upper Hessenberg matrix form ( which costs ) with a finite sequence of orthogonal similarity transforms.
Over general inner product spaces, this is known as the Householder operator.
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 Hermitian transpose of the vector )
:
The Householder matrix has the following properties:
★ it is symmetric:
★ it is orthogonal:
★ therefore it is also involutary: .
Furthermore, really reflects a point (which we will identify with its position vector ) as described above, since
:
where denotes the dot product. Note that is equal to the distance from ''X'' to the hyperplane.
Main articles: 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.
★ Alston S. Householder, Unitary Triangularization of a Nonsymmetric Matrix, ''Journal ACM'', '5' (4), 1958, 339-342. DOI:10.1145/320941.320947
★ David D. Morrison, Remarks on the Unitary Triangularization of a Nonsymmetric Matrix, ''Journal ACM'', '7' (2), 1960, 185-186. DOI:10.1145/321021.321030
★ Householder's Method
★ Householder Transformations
The Householder transformation was introduced in 1958 by Alston Scott Householder.
It can be used to obtain a QR decomposition as described in the QR algorithm of a matrix, and to bring a matrix A to upper Hessenberg matrix form ( which costs ) with a finite sequence of orthogonal similarity transforms.
Over general inner product spaces, this is known as the Householder operator.
| Contents |
| Definition and properties |
| Application |
| References |
| External links |
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 Hermitian transpose of the vector )
:
The Householder matrix has the following properties:
★ it is symmetric:
★ it is orthogonal:
★ therefore it is also involutary: .
Furthermore, really reflects a point (which we will identify with its position vector ) as described above, since
:
where denotes the dot product. Note that is equal to the distance from ''X'' to the hyperplane.
Application
Main articles: 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.
References
★ Alston S. Householder, Unitary Triangularization of a Nonsymmetric Matrix, ''Journal ACM'', '5' (4), 1958, 339-342. DOI:10.1145/320941.320947
★ David D. Morrison, Remarks on the Unitary Triangularization of a Nonsymmetric Matrix, ''Journal ACM'', '7' (2), 1960, 185-186. DOI:10.1145/321021.321030
External links
★ Householder's Method
★ Householder Transformations
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