武汉大学数学与统计学院：《数值分析》第9章 矩阵特征值问题的数值方法（9.1-9.4）特征值和Jacobi方法

第9章矩阵特征值问题的数值方法 91特征值与特征向量 92 Hermite矩阵特征值问题 93 Jacobi方法 94对分法 9.5乘幂法 9.6反幂法 97QR方法

第9章 矩阵特征值问题的数值方法 9.1 特征值与特征向量 9.2 Hermite矩阵特征值问题 9.3 Jacobi方法 9.4 对分法 9.5 乘幂法 9.6 反幂法 9.7 QR方法

引言 工程实践中有多种振动问题,如桥梁或建 筑物的振动,机械机件、飞机机翼的振 动,工程实践中有多种振动问题,如桥 梁或建筑物的振动,机械机件、飞机机 翼的振动,及一些稳定性分析和相关分 析可转化为求矩阵特征值与特征向量的 问题

引言 工程实践中有多种振动问题，如桥梁或建 筑物的振动，机械机件、飞机机翼的振 动，工程实践中有多种振动问题，如桥 梁或建筑物的振动，机械机件、飞机机 翼的振动，及一些稳定性分析和相关分 析可转化为求矩阵特征值与特征向量的 问题

通更 London, England: Millennium(Wobbly')Bridge (1998-2002, Norman Foster and Partners and Arup Associates)

London, England: Millennium ('Wobbly') Bridge (1998-2002, Norman Foster and Partners and Arup Associates)

i decide that i have to write something today otherwise I would not know how to speak english here This is a very quick story about a bridge London launched three major construction projects to celebrate the arrival of the millennium After all Greenwich(pronounced green-ich) is supposed to be (supposed to be? where the prime meridian lies, and the place where the Millennium officially starts in the world. The three projects are the Millennium Dome in North greenwich, so far the largest single roofed structure in the world, London Eye right across Westminster. which becomes so far the largest observation wheel in the world and the millennium Bridge that links southeast london with St. Pauls Cathedral, which is currently..well. not swinging any more. it is said

I decide that I have to write something today, otherwise I would not know how to speak English here. • This is a very quick story about a bridge. • London launched three major construction projects to celebrate the arrival of the Millennium. After all, Greenwich (pronounced green-ich) is supposed to be (supposed to be?!) where the prime meridian lies, and the place where the Millennium officially starts in the world. The three projects are the Millennium Dome in North Greenwich, so far the largest single roofed structure in the world, London Eye right across Westminster, which becomes so far the largest observation wheel in the world, and the Millennium Bridge that links Southeast London with St. Paul’s Cathedral, which is currently…well...not swinging any more, it is said

The bridge was designed by imperial college, a college of my former university. On the very first day that the bridge was open to public, there were simply so many people going there to walk from the south bank to St. Paul's that the weight completely exceeded the architect s expectation The slender steel truss bridge began to vibrate with a million people on there. The opening ceremony ended up in an embarrassing vertigo Millennium left londoners a happy adage about swinging bridge, meaning fancy technology that looks good but functions in a funny fashion Am I using too many F's here? Or is it simply because my tongue starts to swing in the same direction when I am writing about this wobbly bridge Next time you visit London, I strongly recommend this place After all, with a little swing, this is a shortcut to dash into st Paul,'s directly from the southeast

• The bridge was designed by Imperial College, a college of my former university. On the very first day that the bridge was open to public, there were simply so many people going there to walk from the south bank to St. Paul’s that the weight completely exceeded the architect’s expectation. • The slender steel truss bridge began to vibrate with a million people on there. The opening ceremony ended up in an embarrassing vertigo. • Millennium left Londoners a happy adage about swinging bridge, meaning fancy technology that looks good but functions in a funny fashion. • Am I using too many F’s here? Or is it simply because my tongue starts to swing in the same direction when I am writing about this wobbly bridge? • Next time you visit London, I strongly recommend this place. After all, with a little swing, this is a shortcut to dash into St. Paul’s directly from the southeast!

搜索引擎 Google 谷歌 GX=X Badu百度 x e YZHQ!搜索 G: Google Matrix “ the world’ s largest matrix computation” soo搜狗 4,300,000,000 X: Page rank(网页级别) vector 7ASK爱问 “TheS25,0000 Eigenvector” 中搜 搜素- OPen⑥OSO搜搜 我们只搜

G x = x T x e = 1 T G: Google Matrix, “the world’s largest matrix computation”. 4,300,000,000 x: PageRank（网页级别） vector “The $25,000,000,000 Eigenvector” 搜索引擎

91特征值与特征向量 设A是n阶矩阵,ⅹ是非零列向量.如果有 数λ存在,满足Ax=Ax(1) 那么,称ⅹ是矩阵A关于特征值λ的特征向

9.1 特征值与特征向量 设A是n阶矩阵，x是非零列向量. 如果有 数λ存在，满足 ， (1) 那么，称x是矩阵A关于特征值λ的特征向 量

如果把(1)式右端写为Ix,那么(1)式又可写 为 (-A)x=0 即|AI-A|=0 f()=九-A2”+an1”+…+a1元+ao 它是关于参数入的n次多项式,称为矩阵A的特 征多项式,其中a0=(1)n|A

如果把(1)式右端写为 ，那么(1)式又可写 为: Ix ( ) 0 I A x − = 即| | 0 I A− = 1 1 1 0 ( ) | | ... n n n f I A a a a      − = − = + + + + − 记 它是关于参数λ的n次多项式，称为矩阵A的特 征多项式， 其中a0=(-1)n｜A｜. (2)

显然,当λ是A的一个特征值时,它必然 是f()=0的根反之,如果λ是f(4)=的根 那么齐次方程组(2有非零解向量x,使(1)式 成立从而,λ是A的一个特征值 A的特征值也称为A的特征根

显然，当λ是A的一个特征值时，它必然 是 的根. 反之，如果λ是 的根， 那么齐次方程组(2)有非零解向量x，使(1)式 成立. 从而，λ是A的一个特征值. A的特征值也称为A的特征根. f ( ) 0  = f ( ) 0  =

矩阵特征值和特征向量有如下主要性质 定理91.1n阶矩阵A是降秩矩阵的充分必 要条件是A有零特征值 定理91.2设矩阵A与矩阵B相似,那么它们 有相同的特征值. 定理91.3n阶矩阵A与A有相同的特征值. 定理914设λλ是n阶矩阵A的两个互异特征 值,X、y分别是其相应的右特征向量和左特征 向量,那么,Xy=0

矩阵特征值和特征向量有如下主要性质： 定理9.1.1 n阶矩阵A是降秩矩阵的充分必 要条件是A有零特征值. 定理9.1.2 设矩阵A与矩阵B相似，那么它们 有相同的特征值. 定理9.1.3 n阶矩阵A与AT有相同的特征值. 定理9.1.4 设λi≠λj是n阶矩阵A的两个互异特征 值，x、y分别是其相应的右特征向量和左特征 向量，那么，x Ty=0