Householder Transformation - Definition and Properties

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

Famous quotes containing the words definition and/or properties:

    Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.
    Walter Pater (1839–1894)

    A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.
    Ralph Waldo Emerson (1803–1882)