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1、Ch5矩阵特征值与特征向量的计算1.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，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。London,England:Millennium('Wobbly')Bridge(1998-2002,NormanFosterandPartnersandArupAssociates)IdecidethatIhavetowritesomethingtoday,otherwiseIwouldnotknowhowtospeakEn
2、glishhere.Thisisaveryquickstoryaboutabridge.LondonlaunchedthreemajorconstructionprojectstocelebratethearrivaloftheMillennium.Afterall,Greenwich(pronouncedgreen-ich)issupposedtobe(supposedtobe?!)wheretheprimemeridianlies,andtheplacewheretheMillenniumoffic
3、iallystartsintheworld.ThethreeprojectsaretheMillenniumDomeinNorthGreenwich,sofarthelargestsingleroofedstructureintheworld,LondonEyerightacrossWestminster,whichbecomessofarthelargestobservationwheelintheworld,andtheMillenniumBridgethatlinksSoutheastLondon
4、withSt.Paul’sCathedral,whichiscurrently…well...notswinginganymore,itissaid.ThebridgewasdesignedbyImperialCollege,acollegeofmyformeruniversity.Ontheveryfirstdaythatthebridgewasopentopublic,thereweresimplysomanypeoplegoingtheretowalkfromthesouthbanktoSt.Pa
5、ul’sthattheweightcompletelyexceededthearchitect’sexpectation.Theslendersteeltrussbridgebegantovibratewithamillionpeopleonthere.Theopeningceremonyendedupinanembarrassingvertigo.MillenniumleftLondonersahappyadageaboutswingingbridge,meaningfancytechnologyth
6、atlooksgoodbutfunctionsinafunnyfashion.AmIusingtoomanyF’shere?OrisitsimplybecausemytonguestartstoswinginthesamedirectionwhenIamwritingaboutthiswobblybridge?NexttimeyouvisitLondon,Istronglyrecommendthisplace.Afterall,withalittleswing,thisisashortcuttodash
7、intoSt.Paul’sdirectlyfromthesoutheast!G:GoogleMatrix,“theworld’slargestmatrixcomputation”.4,300,000,000x:PageRank（网页级别）vector“The$25,000,000,000Eigenvector”搜索引擎2.幂法设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.幂法就是一种求矩阵按模最大特征值的方法.幂法要求A有完备的特征向量系。即A有
8、n个线性无关的特征向量。在实践中，常遇到的实对称矩阵和特征值互不相同的矩阵就具有这种性质。设A的特征值和特征向量如下：特征值：特征向量：幂法可以求，基本思想很简单.设线性无关，取初值，作迭代设：则有：利用则k足够大时，有可见几乎仅差一个常数所以任意分量相除特征向量乘以任意数，仍是特征向量于是，可得到算法：1.给出初值，计算序列2.若序列相邻两个向量各个分量比趋向于常数，则求矩阵A的按模最大的特征值解取x(0)=(1,0)T,计算x(k)=A
