分块矩阵及其应用

1分块矩阵及其运用摘要分块矩阵是高等代数中的一个重要内容，是处理阶数较高的矩阵时常采用的技巧，也是数学在多领域的研究工具。对矩阵进行适当分块，可使高阶矩阵的运算可以转化为低阶矩阵的运算，同时也使原矩阵的结构显得简单而清晰，从而能够大大简化运算步骤，或给矩阵的理论推导带来方便。有不少数学问题利用分块矩阵来处理或证明，将显得简洁、明快。本文先介绍了分块矩阵的概念、运算，几类特殊的分块矩阵，讨论了分块矩阵的初等变换，接着介绍了分块初等矩阵及其性质，最后分类举例说明了分块矩阵在高等代数中的一些应用，包括在在行列式计算中的应用，在证明矩阵秩的问题中的应用，在矩阵求逆问题中的应用，在解线性方程组问题中的应用，在线性相关性及矩阵分解中的应用，在特征值问题中的应用，在相似与合同问题中的应用以及在其他问题中的应用等。大量的例体现了矩阵分块法的基本思想，说明了应用分块矩阵可以使高等代数中的很多计算与证明问题简单化，所以了解分析并掌握分块矩阵的性质与应用及相关的技巧是非常必要的。关键词矩阵分块矩阵初等变换应用2BlockMatrixanditsApplicationAbstract:Matrixisanimportantconceptinhighalgebra，it'softenusedtodealwithhighordermatrixandit'saninstrumentofmathinmanyfields.Dividingmatrixinaproperwaycanturntheoperationofhighordermatrixintotheoperationofalowordermatrix.Atthesametime，itmakesthestructureoftheoriginalmatrixlooksimpleandclear，soitcansimplifythestepsoftheoperationalotorbringtheconvenienceforthetheoryderivationofmatrix.Alotofmathproblemssolvedorprovedbyusingblockmatrixappearsconcise.Atthebeginning，thispaperintroducestheconceptsandoperationsofblockmatrixandsomespecialkindsofblockmatrix，then，itdiscussestheelementarytransformationofblockmatrixandintroducestheelementaryblockmatrixandit'snatures.Atlast，itexplainstheuseofblockmatrixinhighalgebrabymakingexamplesinseveralkinds，includingtheuseinthecalculationofdeterminant，thetestifyoftheproblemoftherankofmatrix，theansweroftheinverseofmatrix，theanswerofsystemoflinearequations，thelinearcorrelationandthedividingofmatrix，theproblemoftheeigenvalue，thesimilarmatrixandContractmatrixandsoon.Alotofexampleshowsthebasictheoryofblockmatrix，Itshowsthatusingblockmatrixcanmakethecalculationandthetestifyinhighalgebraeasier.Itisnecessarythatwemustlearnandanalyseandgrasptheskillofblockmatrixwhichisanimportantconceptinhighalgebra.Keywords:matrixblockmatrixelementarytransformationapplication3目录1前言.........................................................................................................................12分块矩阵..................................................................................................................12.1分块矩阵的定义....................................................................................................12.2分块矩阵的运算....................................................................................................22.2.1加法...................................................................................................................22.2.2数乘...................................................................................................................22.2.3乘法...................................................................................................................22.2.4转置...................................................................................................................42.3两种特殊的分块矩阵............................................................................................42.3.1分块对角矩阵....................................................................................................42.3.2分块上（下）三角形矩阵..................................................................................52.4两种常见的分块方法............................................................................................62.5分块矩阵的初等变换............................................................................................72.6分块初等矩阵及其性质........................................................................................73分块矩阵的应用.....................................................................................................83.1在行列式计算中的应用........................................................................................93.2在证明矩阵秩的问题中的应用............................................................................173.3在逆矩阵问题中的应用.......................................................................................253.3.1解线性方程组法...............................................................................................263.3.2初等变换法......................................................................................................273.3.3三角分解法......................................................................................................293.4在解线性方程组问题中的应用............................................................................303.4.1齐次线性方程组...............................................................................................303.4.2非齐次线性方程组............................................................................................313.5在线性相关性及矩阵分解中的应用.....................................................................343.5.1关于矩阵列（行）向量的线性相关性...............................................................343.5.2矩阵的分解......................................................................................................3443.6在特征值问题中的应用......................................................................................353.7分块矩阵在相似问题中的应用...........................................................................373.8分块矩阵在合同问题中的应用...........................................................................383.9