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Householder reflection ：ウィキペディア英語版

Householder transformation
In linear algebra, a Householder transformation (also known as Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm. The Householder transformation was introduced in 1958 by Alston Scott Householder.
Its analogue over general inner product spaces is the Householder operator.
==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:
:

x - 2\langle x,v\rangle v = x - 2 v (v^\text x),

where ''v'' is given as a column unit vector with Hermitian transpose ''v''^H. This is a linear transformation given by the Householder matrix:
:

P = I - 2 v v^\text\,

, where ''I'' is the identity matrix.
The Householder matrix has the following properties:
* it is Hermitian:

P = P^\text,

* it is unitary:

P^=P^\text,

* hence it is involutory:

P^2=I

.
* A Householder matrix has eigenvalues

\pm 1

. To see this, notice that if

u

is orthogonal to the vector

v

which was used to create the reflector, then

Pu = u

, i.e., 1 is an eigenvalue of multiplicity

n-1

, since there are

n-1

independent vectors orthogonal to

v

. Also, notice

Pv = -v

, 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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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