Definition and Properties
The reflection hyperplane can be defined by a unit vector v (a vector with length 1) which is orthogonal to the hyperplane. The reflection of a point x about this hyperplane is:
where v is given as a column unit vector with Hermitian transpose vH. This is a linear transformation given by the Householder matrix:
- , where I is the identity matrix.
The Householder matrix has the following properties:
- it is Hermitian:
- it is unitary:
- hence it is involutary: .
- 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., 1 is an eigenvalue of multiplicity, since there are independent vectors orthogonal to . Also, notice, and so -1 is an eigenvalue with multiplicity 1.
- The determinant of a Householder reflector is -1, since the determinant of a matrix is the product of its eigenvalues.
Read more about this topic: Householder Transformation
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