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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Householder transformation ： 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: :$x\; -\; 2\backslash langle\; x,v\backslash rangle\; v\; =\; x\; -\; 2\; v\; (v^\backslash text\; x),$ where ''v'' is given as a column unit vector with Hermitian transpose ''v''H. This is a linear transformation given by the Householder matrix: : $P\; =\; I\; -\; 2\; v\; v^\backslash text\backslash ,$, where ''I'' is the identity matrix. The Householder matrix has the following properties: * it is Hermitian: $P\; =\; P^\backslash text,$ * it is unitary: $P^=P^\backslash text,$ * hence it is involutory: $P^2=I$. * A Householder matrix has eigenvalues $\backslash pm\; 1$. 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）』 ■ウィキペディアで「Householder transformation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.