特征值与特征向量及其应用

I摘要特征值与特征向量是代数中一个重要的部分，并在理论和学习和实际生活，特别是现代科学技术方面都有很重要的作用.本文主要讨论并归纳了特征值与特征向量的性质，通过实例展现特征值与特征向量的优越性，探讨特征值与特征向量及其应用有着非常重要的价值.正文共分四章来写，其中第一章介绍了写作背景以及研究目的.第二章介绍了特征值与特征向量的定义以及性质，并且写出了线性空间中线性变换的特征值、特征向量与矩阵的特征值、特征向量之间的关系.第三章介绍了特征值与特征向量的几种解法：利用特征方程求特征值进而求特征向量、列行互逆变换法、利用矩阵的初等变换求特征值和特征向量.第四章重点介绍了特征值特征向量的应用，如n阶矩阵的高次幂的求解以及矩阵特征值反问题的求解等等.本文充分利用特征值与特征向量的特性求解相关问题，这带有一定的技巧性，但并不难想象，特别是跟其它方法相比，计算显得非常简洁，在解决具体问题上具有很大的优越性.当然关于矩阵的特征值和特征向量的内容很广，本文仅就特征向量的性质以及一些应用展开研究.关键词：特征值；特征向量；矩阵；递推关系；初等变换IIAbstractAsanimportantpartofalgebra,EigenvalueandEigenvectorofaMatrixhaveveryimportantapplicationsintheoreticalstudyandpracticallife,especiallyinmodernscienceandtechnology.Inthispaper,somepropertiesofeigenvalueandeigenvectorarediscussedandsummarized,itshowsthesuperiorityofeigenvalueandeigenvectorthroughexamples.Ithasaveryimportantvalueofexploringeigenvalueandeigenvectoranditsapplication.Thetextisdividedintofourchapterstowrite,Amongthem,thefirstchapterpresentsthebackgroundandresearchpurposes.Thesecondchapterpresentsthedefinitionofeigenvalueandeigenvectorandtheirproperties,itwritestherelationshipbetweentheeigenvalue,eigenvectorofthelineartransformofthelinearspaceandeigenvaluesandeigenvectorsofmatrix.Thethirdchapterpresentsseveralsolutionsoftheeigenvalueandeigenvector:thecharacteristicequationforeigenvalueandeigenvector;themethodofreversibletransformonRowsandcolumns;theelementarytransformationofmatrixinverseforeigenvaluesandeigenvectors.Thefourthchapterintroducestheapplicationofeigenvalueeigenvector,suchassolvingthehighpowerofnordermatrix,dealingwiththeinverseproblemofmatrixeigenvaluesandetc.Thispaperfullyutilizeeigenvalueandeigenvectortosolverelatedissues,thisapproachneedscertainskills，butitisnothardtoimaginethatithasthegreatsuperiorityinsovlingspecificissues,comparingwithothermethods.Ofcourse,thecontentaboutmatrixeigenvaluesandeigenvectorsisverywide,thisarticlemainlydealswiththepropertiesofeigenvectorandsomeapplication.Keywords:eigenvalue；eigenvector；matrix；recursiverelations；elementary；transformation目录摘要.......................................................................................................................................IAbstract...................................................................................................................................II1引言..................................................................................................................................11.1研究背景....................................................................................................................11.2研究现状....................................................................................................................11.3本文研究目的及意义................................................................................................22特征值与特征向量..............................................................................................................32.1特征值与特征向量的定义和性质............................................................................32.1.1线性变换的特征值与特征向量......................................................................32.1.2n阶方阵的特征值与特征向量.....................................................................32.2,Vpn中线性变换的特征值、特征向量与矩阵R的特征值与特征向量之间的关系...............................................................................................................................33特征值与特征向量的解法..................................................................................................53.1求数字方阵的特征值与特征向量............................................................................53.2列行互逆变换法........................................................................................................63.3利用矩阵的初等变换解特征值特征向量..............................................................104矩阵的特征值与特征向量的应用研究............................................................................154.1n阶矩阵1*,,,,,mkAaAbIAAAfA的特征值和特征向量...........................154.2n阶矩阵的高次幂的求解.......................................................................................164.3矩阵特征值反问题的求解......................................................................................174.4特征值与特征向量在线性递推关系中的应用......................................................184.5特征值法求解二次型的条件最值问题..................................................................224.5.1二次型的条件最值问题及求解该问题的特征值方法................................224.5.2应用举例........................................................................................................254.6特征值与特征向量在矩阵运算中的作用..............................................................264.6.1特征值与特征向量在矩阵运算中使用的性质............................................264.6.2特征值与特征向量在矩阵运算中的应用....................................................26总结....................................................................................................................................30参考文献................................................................................................................................31致谢........................................................