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Concept
Householder transformation
Résumé
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.
Its analogue over general inner product spaces is the Householder operator.
The reflection hyperplane can be defined by its normal vector, a unit vector (a vector with length ) that is orthogonal to the hyperplane. The reflection of a point about this hyperplane is the linear transformation:
where is given as a column unit vector with conjugate transpose .
The matrix constructed from this transformation can be expressed in terms of an outer product as:
is known as the Householder matrix, where is the identity matrix.
The Householder matrix has the following properties:
it is Hermitian: ,
it is unitary: ,
hence it is involutory: .
A Householder matrix has eigenvalues . To see this, notice that if is orthogonal to the vector which was used to create the reflector, then , i.e., is an eigenvalue of multiplicity , since there are independent vectors orthogonal to . Also, notice , and so is an eigenvalue with multiplicity .
The determinant of a Householder reflector is , since the determinant of a matrix is the product of its eigenvalues, in this case one of which is with the remainder being (as in the previous point).
In geometric optics, specular reflection can be expressed in terms of the Householder matrix (see ).
Householder transformations are widely used in numerical linear algebra, for example, to annihilate the entries below the main diagonal of a matrix, to perform QR decompositions and in the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form. For symmetric or Hermitian matrices, the symmetry can be preserved, resulting in tridiagonalization.
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