4.2.1 Householder矩阵 1.定义如下形式的：阶方阵。 H=I,-2uuT,MeR且aFh=1(或w=1)+ 称为初等反射裤（或锁象麦换裤，或Househol der矩车）；由初等反射阵H确定 的R”的变换y=c称为初等反射变换或Househol der变换。· This documentis produced bytril versinofPrVisit www.prinashcmformore infomio
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2.性质设H是初等反射阵，由定义容易证明H的如下一些性质： 1)HT-H(实对称阵)： H=(I,-2uu')=I.-2uu=H 证毕 2)HTH=I(正阵)： HH=(I -2uu)2=I -2uu-2mu +4un'unT=I 证毕 3)H=I.(对合阵)：4)H=H(自逆阵)： 5)detH=-1.4 由上节所证的结果，有dtH=det(I,-2w')=1-2mw=1-2=-1。证毕 This documentis prduced byriaversion ofPrnh Visit www.prinashcfor more informion
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3.几何解释. 在R3中说明称之为初等反射阵的原因。考虑以为法向量且过原点的平面刀 (见图)。 任取x∈R3,将x分解为 x=+w,其中y,wL 则，='v=0(正交)，w=(共线)。从而 Hx =(-2uuT)x=x-2uux=x-2mu (v+w)= =x-2uuw=v+w-2uu ()=v+w-2u=v-w=x' 可见H作用于向量x后，将其关于以为法向量的平面π反射变为x'。 This documentis produced bytril versinofPrVisit www.prinashmformore infomio
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4.一些重要结论 定理设H是阶初等反胖，则O）是+r阶初等反胖。 o 证因为H=1,-2w,且u=1,所以 -68）-9听 其申马-但配的4=。Q--1伯)初等反纳降 This documentis prduced byria version ofPrnhVisit www.prinashcfor more informion
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定理设：∈R"是给定的单位向星，即kl,=1,则对任意xeR,达存在初 等反射阵H,使得 孤=az,其中a=r2 证若x=0,任职单位向量M,则H=I-2umF满足=0=az,成立。 若x=xz,取满足(x,)=n”x=0的单位向量w,则=x=z,成立。 若x4,取“则是羊位向蛋，且有 x-cz e-&z6 -2x-a2-+c-aa=xx- x"x-az"x 2(xx-azx) =x-(x-c)=z 证毕 This documentis produced bytril versinofPrVisit www.prinashcmformore infomio
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