# 豪斯霍尔德变换

（重定向自Householder变换

## 定义

${\displaystyle H=I-2vv^{*}.\,}$

## 性质

• 它是對稱矩陣，即 ${\displaystyle H^{T}=H}$
• 它是正交矩阵，即 ${\displaystyle H^{T}=H^{-1}}$
• 它是埃爾米特矩陣，即 ${\displaystyle H^{*}\ =H}$
• 它是对合的，即 ${\displaystyle H^{2}=I\ }$

${\displaystyle Hx=x-2vv^{*}x=x-2\langle v,x\rangle v,}$

## 应用

${\displaystyle \mathbf {H} =\mathbf {I} -{\frac {2}{\langle \mathbf {v} ,\mathbf {v} \rangle }}\mathbf {vv} ^{H}}$

${\displaystyle \mathbf {v} =\mathbf {x} +{\rm {{sgn}(x_{1})\Vert x\Vert _{2}\mathbf {e} _{1}.\,}}}$

Dubrulle 在2000年给出了将豪斯霍尔德变换应用于生成一个一般的稀疏向量的一个数值稳定的算法[4]

## 参考文献

1. ^
2. ^ H.W. Turnbull, A.C. Aitken, An Introduction to the Theory of Canonical Matrices, Blackie, London: Glasgrow, 1932
3. ^ Alston S. Householder, Unitary Triangularization of a Nonsymmetric Matrix, Journal ACM, 5 (4), 1958, 339-342. DOI:10.1145/320941.320947
4. ^ A.A. Dubrulle, Householder Transformations Revisited, SIAM Journal on Matrix Analysis and Applications, 2001
5. ^ David D. Morrison, Remarks on the Unitary Triangularization of a Nonsymmetric Matrix, Journal ACM, 7 (2), 1960, 185-186. DOI:10.1145/321021.321030