本科毕业论文-浅谈分块矩阵的运算合理性及其在求逆矩阵中的运用

分类号01512论文编号201040432023本科生毕业论文浅谈分块矩阵的运算合理性及其在求逆矩阵中的运用姓名院系数学科学学院年级专业2010级数学与应用数学指导教师张裕波2014年5月目录摘要IIABSTRACTIII第一章绪论1第二章分块矩阵的概念及其运算合理性121分块矩阵的概念122分块矩阵的运算合理性2221分块矩阵乘法的合理性2222分块矩阵的初等变换及其合理性3第三章分块矩阵的运算在求逆矩阵中的运用531利用分块矩阵的思想探讨逆矩阵的计算问题532求分块矩阵的逆矩阵8第四章结论9参考文献10致谢11摘要分块矩阵乘法的合理性是指,在作矩阵乘法时,对矩阵的块可以像对矩阵的元素一样对待分块矩阵的初等变换与矩阵乘法有十分密切的关系,直接依赖于所涉及到的矩阵乘法是否能进行分块矩阵的三类初等变换,其实就是对不分块的原矩阵进行若干次初等变换有趣的是,分块矩阵的初等变换,比普通的初等变换,其“规模”要大得多,是一种“成批量”的初等变换探讨逆矩阵的求法问题,是分块矩阵运算的典型运用求逆矩阵的典型例子,反映了分块矩阵的乘法运算与初等变换之间的深刻联系关键词分块矩阵初等变换合理性矩阵求逆ABSTRACTBLOCKMATRIXMULTIPLICATIONRATIONALITYREFERSTO,INTHEMATRIXMULTIPLICATION,THEBLOCKMATRIXCANBETREATEDASTHEMATRIXELEMENTSTHEREISVERYCLOSERELATIONSHIPBETWEENTHEELEMENTARYTRANSFORMATIONOFBLOCKMATRIXANDMATRIXMULTIPLICATION,MATRIXMULTIPLICATIONDIRECTLYDEPENDSONWHETHERTHECANTHREEKINDSOFELEMENTARYTRANSFORMATIONOFBLOCKMATRIX,ISACTUALLYTOTHEORIGINALMATRIXDOESNOTBLOCKARESEVERALELEMENTARYTRANSFORMATIONINTERESTINGLY,THEELEMENTARYTRANSFORMATIONOFBLOCKMATRIX,ELEMENTARYTRANSFORMATIONTHANORDINARY,ITS“SIZE“ISMUCHGREATER,ISAKINDOF“ELEMENTARYTRANSFORMATIONINTOTHEBULK“TOEXPLORETHEMETHODOFINVERSEMATRIX,ISATYPICALUSINGBLOCKMATRIXOPERATIONSATYPICALEXAMPLEOFTHEINVERSEMATRIX,REFLECTSTHEPROFOUNDRELATIONBETWEENTHEMULTIPLICATIONANDTHEELEMENTARYTRANSFORMATIONOFBLOCKMATRIXKEYWORDSBLOCKMATRIXELEMENTARYTRANSFORMATIONRATIONALITYMATRIXINVERSION第一章绪论矩阵是数学中一个重要的基本概念,是代数学中重要的研究对象之一,是数学研究和应用的重要工具,矩阵的性质依赖于矩阵中元素的性质,矩阵由最初的一种工具经过近两个世纪的发展,形成了一门较为完善的学科矩阵论矩阵理论应用于许多领域,其中在数学和工程学中应用最为广泛在数学上,矩阵是求解线性方程组,研究线性空间,线性变换等问题的最要工具,而在一些阶数较高的矩阵中，计算量却是相当复杂，此时引出分块矩阵，分块矩阵也是矩阵论中一个重要的概念,在线性代数及高等代数中扮演着不可缺少的角色,分块矩阵在进行行列式的计算,求解线性方程组,特别是在求逆矩阵的问题上,有非常重要的作用矩阵分块,就是将一个高阶数的矩阵看成是由一些低阶数的矩阵组成的,从而将这些低阶数的矩阵看成是高阶数的矩阵的元素,特别是在矩阵的相关运算中,把这些小矩阵看成是数来处理,为矩阵的运算带来了许多方便,将矩阵分块之后,矩阵之间的相互关系,可以看得十分清楚,能够在解决实际问题中,使问题变得简单化本文即是通过查阅相关文献和学习相关知识后总结和探究分块矩阵的运算的合理性以及分块矩阵在求逆矩阵方面的应用通过对分块矩阵的运算合理性及分块矩阵的初等变换的研究,总结和探究分块矩阵在求逆矩阵方面的应用,从而得出分块矩阵在处理很多问题上带来方便第二章分块矩阵的概念及其运算合理性21分块矩阵的概念定义1在矩阵的行或列之间划上横线或竖线将其分为若干部分，称为对矩阵进行分块,分了块的矩阵称为分块矩阵显然,对于一个的矩阵,最多可以分成块（一个元素一块）,最少MNMN可以分成一块（它本身）注1、在对矩阵进行分块的过程中,必须注意分块的“线”必须穿过整个矩阵且必须是直线，如果在行或列之间所划直线没有穿过矩阵，或者所划的线不是直线而是“折线”，这种分块分法均是错误的错误的分块方法如（线条未穿过）123405A2、矩阵的分块要根据需要选择正确的分法，应有利于认识矩阵的结构，便于矩阵的运算和讨论有关矩阵的性质22分块矩阵的运算合理性普通矩阵可以进行加（含减）乘（含除和乘方）以及数乘等运算因为矩阵的转置、矩阵的初等变换与矩阵的运算有密切联系,所以本文也将转置、初等变换视为矩阵的运算,分块矩阵同样可以进行加、乘、数乘、转置及初等变换等运算，并且限于矩阵的运算结果而言，在对矩阵进行正确分块的基础上，通过“块”运算得出的结果与不分块直接运算得出的结果是一致的对此,我们仅就分块矩阵的乘法和初等变换加以说明221分块矩阵乘法的合理性分块矩阵乘法的合理性是指,在作矩阵乘法时,对矩阵的块可以像对矩阵的元素一样对待即先对相乘的两个矩阵作正确的分块（后一个矩阵的行的分法与前一个矩阵的列的分法一致）,然后把“块”看成元素,把分块矩阵的乘法按照普通矩阵的乘法进行（但要严格遵守块的前后顺序）,得到的积是由一些“乘积块”构成的新的分块矩阵,再将每一“乘积块”按照普通矩阵乘法乘出来,“安放”在相应的位置,从而得到原来两个矩阵的乘积这种分了块再行计算与不分块直接计算,其结果是一致的设，则有,MNIJIJSNNAABBP12121121211212TVVTUUTTTVUUVABCSN其中；21,12,PQQPQPTQPKQCBAQ下面说明分块矩阵乘法的合理性计算中元设C,IJ,IJ，其中,121PSSF0PFS，其中,QJMGQGM即是设中元素位于中位置矩阵的第行属于第个行C,IJPC（F,AIP组,矩阵的第列属于的第个列组则有以下式子成立BB1,TPQPKQIJFGAFG11LNTTPKQPKKQKRBFFBRG211211TNNNPQPQPTQRRRAFGAFAFBRG212111TTNNNRRRIBJIBJIJ1,NRAIJAIJ222分块矩阵的初等变换及其合理性我们知道,分块矩阵的初等变换与普通矩阵的初等变换类似,包含三类1互换分块矩阵的两块行或两块列；2以某个可逆矩阵左乘以分块矩阵的某一块行，或右乘以某一块列；3以某个矩阵左乘以分块矩阵的某一块行后加到另一块行上去，或以某个矩阵右乘以分块矩阵的某一块列后加到另一块列上去（这里假定上面提到的运算都是可以进行的）与普通矩阵一样,对于分块矩阵,也有分块初等矩阵的概念显然,分块矩阵的初等变换与矩阵乘法有十分密切的关系,直接依赖于所涉及到的矩阵乘法是否能进行,即是说,不是任意矩阵（即使可逆）都可以乘到分块矩阵的某一块行或块列上即使可以乘,左乘与右乘也有严格的要求对于分块矩阵的初等变换的合理性,我们主要说明将一个矩阵分了块以后进行块的初等变换,其结果就是对原矩阵进行若干次普通初等变换对此,设12SMABCDN1互换的两块行（列），显然相当于多次互换原矩阵的行（列）2设可逆，则可以分解成初等矩阵的乘积，在乘法可以PLP21进行的情况下,将左乘以的第一块行A12LLPABABCDD容易看出,其结果相当于对的前行作次初等行变换（列的情况的说S1ML明与此类似）3设，121312321231233SABABCDCD123PP我们来考察，PAB通过计算，得CD11213112132222AAABBBCPCPCPDPDPDP313123123131313213231一般情况的证明与此完全类似,只是符号较多而已,（列变换的证明也与此类似）由此看来,分块矩阵的三类初等变换,其实就是对不分块的原矩阵进行若干次初等变换第一类分块初等变换就是互换矩阵的行（列）,第二类分块初等变换限于对所乘可逆矩阵的那些块行（块列）内作初等变换,第三类分块初等变换涉及到其它块行（块列）有趣的是,分块矩阵的初等变换,比普通的初等变换,其“规模”要大得多,是一种“成批量”的初等变换因此,分块初等矩阵、分块矩阵的初等变换与普通的初等矩阵、初等变换具有类似的性质分块初等矩阵是可逆矩阵,其逆矩阵仍是初等分块矩阵；矩阵的分块行（列）初等变换相当于同类分块初等矩阵左（右）乘以被变换的矩阵；进行分块初等变换不改变矩阵的秩,如果是方阵,则不改变它的行列式是否为零的属性（第一类分块矩阵初等变换不改变行列式的绝对值,第二类分块矩阵初等变换不改变行列式的值）第三章分块矩阵的运算在求逆矩阵中的运用分块矩阵的运算在数学中有广泛的运用，这里只揭示分块矩阵的运算在求逆矩阵方面的运用31利用分块矩阵的思想探讨逆矩阵的计算问题设可逆,则可用初等行变换求的逆,也可用初等列变换求的逆,还可以AAA同时使用行、列初等变换来求1、行变换的一般方法为把用初等行变换化为单位矩阵,同时对用E相同的行变换即可化得事实上，设可逆,则有初等矩阵,使得112TP,由此可得,T12PAET12T12AP格式为EA行变换行变换用等式表示为1T12PAEPP（这里，设,则当式中时,）T12AE2、列变换的一般方法为把用初等列变换化为单位矩阵,同时对用E相同的列变换即可化得事实上，设可逆,则有初等矩阵,使得1A12TQ,此时,12TAQE1212TTQE格式为1A列变换列变换用等式表示为121TAAQEQEEA（这里，设,则当式中时,）12,T1QA3、可以把行变换和列变换结合在一起,同时使用行变换和列变换求即1取,其中均为初等矩阵,构造分块矩T12P12,SQIJP阵并作运算OAEAEPQO从式可以看出,可以对进行一系列的初等变换,当块的位置即A时,PAQE1P例1设,用初等变换求矩阵的逆215A1解I初等行变换的方法012012101250AE010431013014130AII初等列变换的方法12101010105203431101211000AE1430AIII同时使用行变换和列变换的方法1201015201001AEO01201201001100113322010001EPQO1243101030AQP32求分块矩阵的逆矩阵例2设是一个分块矩阵,其中为阶方阵,为阶方阵,6ABPCDBRCK当与都是可逆矩阵时,求证可逆,并求B1P1解法一因为与都是可逆矩阵,B1A且11AEOBCDCD1110XBOPBCDAAEA（所以可逆11212121111110000DBRCDBARRRBARAECECDBEA11110ECBCADBD11111PA解法二ADBCOEADBCEODCBA111111111OP11111CDBACDBA第四章结论分块矩阵是高等代数的基本工具,分块矩阵的运算（包括初等变换）是高等代数的基本语言但是分块矩阵的运算（特别是分块矩阵的乘法和初等变换）的合理性并不是一目了然通过本文的讨论,我们看到,求逆矩阵的问题集中反映了分块矩阵的深刻思想内涵分块矩阵的乘法与初等变换之间有着深刻的内在联系参考文献1北京大学数学系几何与代数教研室前代数小组高等代数M第三版北京高等教育出版社,200372姚慕生高等代数学M复旦大学出版社,200533智婕分块矩阵的初等变换J甘肃联合大学学报自然科学版,2011,25922234张新育,周世国矩阵的分块初等变换与分块初等矩阵及其应用J数学的实践与认识,2011,41192452485陈文华分块矩阵的初等变换及其应用J大理学院院报,2009,8876严坤妹分块矩阵的应用J福建广播电视大学学报,2006,571727高百俊分块矩阵的初等变换及应用J伊犁师范学院学报自然科学版,2007,416178王秀芳分块矩阵的应用讨论J连云港师范高等专科学校学报,2008,397989丘维声高等代数上M北京清华大学出版社,20106,19421110王萼芳,石生明高等代数M第三版北京高等教育出版社,200392010重印,181186致谢值此毕业论文完成之际,特感谢我的论文指导老师张裕波老师,他是一位认真负责,热情的老师,在整个毕业论文的撰写过程中,他都对我进行了认真的指导和详细的修改,从他的身上我也学到了很多,比如说严谨认真、一丝不苟的工作作风我将永远记住他曾经给予过我的教导，我要向张老师表达我内心最崇高的敬意和最衷心的感谢同时我还要感谢四年来所有任课老师和辅导员及同学们,在我大学四年的学习和生活中,给予了我很多的教导和很大的帮助，是他们的理解、包容、支持使我逐渐成长，从他们身上我学到了很多做人的道理,将使我受益终生感谢我的家人和好友,他们的关爱和支持永远是我前进的最大动力,在任何时候,他们都给予我最大的鼓励和支持,我感谢他们最后,向审阅我论文和参加答辩的老师们表示衷心的感谢,感谢你们抽出宝贵的时候参加我的论文答辩会,感谢你们对论文不当之处提出的宝贵意见和建议请您删除一下内容，O_O谢谢MANYPEOPLEHAVETHESAMEMIXEDFEELINGSWHENPLANNINGATRIPDURINGGOLDENWEEKWITHHEAPSOFTIME,THESEVENDAYCHINESE请您删除一下内容，O_O谢谢NATIONALDAYHOLIDAYCOULDBETHEBESTOCCASIONTOENJOYADESTINATIONHOWEVER,ITCANALSOBETHEEASIESTWAYTORUINHOWYOUFEELABOUTAPLACEANDYOUMAYBECOMEMOREFATIGUEDAFTERTHEHOLIDAY,DUETOBATTLINGTHELARGECROWDSDURINGPEAKSEASON,ADREAMABOUTAPLACECANTURNTONIGHTMAREWITHOUTCAREFULPLANNING,ESPECIALLYIFYOUTRAVELWITHCHILDRENANDOLDERPEOPLEASMOSTCHINESEPEOPLEWILLTAKETHEHOLIDAYTOVISITDOMESTICTOURISTDESTINATIONS,CROWDSANDBUSYTRAFFICAREINEVITABLEATMOSTPLACESALSOTOBEEXPECTEDAREINCREASINGTRANSPORTANDACCOMMODATIONPRICES,WITHTHEPOSSIBILITYTHATTHEREWILLBENOROOMSAVAILABLEITISALSOCOMMONTHATYOULLWAITINTHELINEFORONEHOURTOGETATICKET,ANDANOTHERTWOHOURSATTHESITE,TOONLYSEEATINYBITOFTHEPLACEDUETOTHECROWDSLASTYEAR,428MILLIONTOURISTSTRAVELEDINCHINAOVERTHEWEEKLONGHOLIDAYINOCTOBERTRAVELINGDURINGTHISPERIODISAMATTERTHATNEEDSTHOROUGHPREPARATIONIFYOUARESHORTONTIMETOPLANTHEUPCOMING“GOLDENWEEK“ITMAYNOTBEABADIDEATOAVOIDSOMEOFTHEMOSTCROWDEDPLACESFORNOWTHEREISALWAYSAPLACESOFASCINATINGTHATEVERYONEYEARNSFORARXANISAPLACELIKETHISTHEBEAUTYOFARXANISEVERLASTINGREGARDLESSOFTHECHANGINGOFFOURSEASONSBESTOWEDBYNATURE,ITSSPECTACULARSEASONALLANDSCAPEANDMOUNTAINSAREJUSTBEYONDWORDARXANISACRUCIALDESTINATIONFORTHERECOMMENDEDTRAVELLINGROUTE,“CHINAINNERMONGOLIAARXANHAILARMANZHOULI“ITISALSOTHEJOINTOFTHEFOURPRAIRIESACROSSTHESINOMONGOLIANBORDER,WHEREPEOPLEGRAVITATETOWARDSTHEEXOTICATMOSPHEREMIXEDWITHCHINESE,RUSSIAN,ANDMONGOLIAELEMENTSASAHISTORICSITEFORTHEYITIANBATTLE,ARXANSTILLEMBODIESTHESPIRITOFGENGHISKHANWALKINGINTOARXAN,YOUWILLBEAMAZEDBYAKALEIDOSCOPEOFGORGEOUSCOLORSALLTHEYEARROUNDTHESPRINGAZALEASBLOOMINGREDINTHESNOW,THESUMMERSEAWAVERINGBLUEINTHEBREEZE,THEAUTUMNLEAVESPAINTEDINYELLOWCOVERINGVOLCANICTRACES,ANDTHEWINTERWOODSSHININGWHITEONTHEVASTALPINESNOWSCAPEHINGGANLEAGUEARXANCITYISSITUATEDINTHEFAREASTERNAREAOFINNERMONGOLIAAUTONOMOUSREGIONITSFULLNAME“HARENARXAN“MEANS“HOTHOLYWATER“INTHEMONGOLIANLANGUAGEARXANISATOURISMCITYINTHENORTHERNFRONTIERWITHABLENDOFLARGEFOREST,GRANDPRAIRIES,VASTSNOWFIELD,HEAVENLAKECLUSTER,THERMIUM,ASWELLASVOLCANICCLUSTERITISARAREANDUNIQUEECOTOURISMBASEFILLEDWITHHEALTHYSUNSHINE,CLEANAIRANDUNSPOILEDGREENNESTLEDCLOSETOTHECOUNTRYSLARGESTVIRGINFOREST,ANDKNOWNFORITSSPRINGANDECOLOGICALENVIRONMENT,ARXANISMARVELEDATBYMANYTOURISTSASTHEPURESTLANDONEARTHYOUCANNOTMISSOUTTHEAUTUMNOFARXANITISDEFINITELYTHEBESTWITHBRIGHTLYCOLOREDSCENERYFULLOFEMOTIONSAUTUMNINTHENORTHERNPARTOFTHECOUNTRYCOMESEARLIERTHANTHESOUTHASEPTEMBERRAINFOLLOWEDBYTHEFOOTPRINTSOFAUTUMNBRINGSMORECOLORSTOTHEONCEEMERALDGREENMOUNTAINANDBLOOMINGGRASSLANDSHUTTERBUGSFLOCKTOSEEFORTHEMSELVESTHEMARVELOFSPLENDIDCOLORSAROUNDTHEMOUNTAINSANDWATERS,MANYOFWHOMHAVETRAVELALONGDISTANCEANDEVENCAMPHEREONLYTOCAPTUREAMOMENTOFTHENATUREWONDERTHESILVERBIRCHTURNSGOLDEN,WHILETHELARCHISSTILLPROUDLYGREENYOUWILLFINDYOURSELFDROWNEDINTHEINTOXICATINGREDOFTHEWILDFRUITSASWELLASTHEGLAMOUROFFLOWERSINFULLBLOWNANDYOURHEARTWILLBELINGERINGONTHEWOODSASITSTIMEFORTHEWILDFRUITSTORIPETHEPICTURESQUEARXANINAUTUMNISINDEEDAFAIRYLANDONLYEXISTSINADREAMTHATSATISFIESALLYOURFANTASIESIFITRAINSHEAVILYONSATURDAYNIGHT,SOMEELDERLYCHINESEWILLSAYITISBECAUSEZHINU,ORTHEWEAVINGMAID,ISCRYINGONTHEDAYSHEMETHERHUSBANDNIULANG,ORTHECOWHERD,ONTHEMILKYWAYMOSTCHINESEREMEMBERBEINGTOLDTHISROMANTICTRAGEDYWHENTHEYWERECHILDRENONQIXI,ORTHESEVENTHNIGHTFESTIVAL,WHICHFALLSONTHESEVENTHDAYOFTHESEVENTHLUNARMONTH,WHICHISUSUALLYINEARLYAUGUSTTHISYEARITFALLSONSATURDAY,AUGUST2FOLKLORESTORYASTHESTORYGOES,ONCETHEREWASACOWHERD,NIULANG,WHOLIVEDWITHHISELDERBROTHERANDSISTERINLAWBUTSHEDISLIKEDANDABUSEDHIM,ANDTHEBOYWASFORCEDTOLEAVEHOMEWITHONLYANOLDCOWFORCOMPANYTHECOW,HOWEVER,WASAFORMERGODWHOHADVIOLATEDIMPERIALRULESANDWASSENTTOEARTHINBOVINEFORMONEDAYTHECOWLEDNIULANGTOALAKEWHEREFAIRIESTOOKABATHONEARTHAMONGTHEMWASZHINU,THEMOSTBEAUTIFULFAIRYANDASKILLEDSEAMSTRESSTHETWOFELLINLOVEATFIRSTSIGHTANDWERESOONMARRIEDTHEYHADASONANDDAUGHTERANDTHEIRHAPPYLIFEWASHELDUPASANEXAMPLEFORHUNDREDSOFYEARSINCHINAYETINTHEEYESOFTHEJADEEMPEROR,THESUPREMEDEITYINTAOISM,MARRIAGEBETWEENAMORTALANDFAIRYWASSTRICTLYFORBIDDENHEORDEREDTHEHEAVENTROOPTOCATCHZHINUBACKNIULANGGREWDESPERATEWHENHEDISCOVEREDZHINUHADBEENTAKENBACKTOHEAVENDRIVENBYNIULANGSMISERY,THECOWTOLDHIMTOTURNITSHIDEINTOAPAIROFSHOESAFTERITDIEDTHEMAGICSHOESWHISKEDNIULANG,WHOCARRIEDHISTWOCHILDRENINBASKETSSTRUNGFROMASHOULDERPOLE,OFFONACHASEAFTERTHEEMPRESSTHEPURSUITENRAGEDTHEEM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