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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)