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

1、分类号 0151.2 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12.1 分块矩阵的概念12.2 分块矩阵的运算合理性22.2.1 分块矩阵乘法的合理性22.2.2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53.1 利用分块矩阵的思想探讨逆矩阵的计算问题53.2 求分块矩阵的逆矩阵8第四章 结论9参考文献10致 谢

2、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AbstractBlock matrix multiplication rationality r

3、efers to, in the matrix multiplication, the block matrix can be treated as the matrix elements. There is very close relationship between the elementary transformation of block matrix and matrix multiplication, matrix multiplication directly depends on whether the can. Three kinds of elementary trans

4、formation of block matrix, is actually to the original matrix does not block are several elementary transformation. Interestingly, the elementary transformation of block matrix, elementary transformation than ordinary, its "size" is much greater, is a kind of "elementary transformatio

5、n into the bulk". To explore the method of inverse matrix, is a typical using block matrix operations. A typical example of the inverse matrix, reflects the profound relation between the multiplication and the elementary transformation of block matrix.Keywords: Block matrix Elementary transform

6、ation rationality Matrix inversion 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

7、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

8、算合理性2.1 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）2、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.2.2 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）

9、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.2.2.1 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

10、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:其中；下面说明分块矩阵乘法的合理性.计算中元.设，其中,，其中,即是设中元素位于中位置.矩阵的第行属于第个行组,矩阵的第列属于的第个列组.则有以下式子成立 = .2.2.2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

11、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(1)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

12、，则可以分解成初等矩阵的乘积:，在乘法可以进行的情况下,将左乘以的第一块行: 容易看出,其结果相当于对的前行作次初等行变换.（列的情况的说明与此类似）(3)设，我们来考察，通过计算，得： 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

13、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.3.1 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,还可以同时使用行、列

14、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,同时对用相同的行变换即可化得.事实上，设可逆,则有初等矩阵,使得,由此可得, 格式为：用等式表示为: （这里，设,则当式中时,.）2、列变换的一般方法为:把用初等列变换化为单位矩阵,同时对用相同的列变换即可化得.事实上，设可逆,则有初等矩阵,使得,此时,格式为:用等式表示为: （这里，设,则当式中时,.）3、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:取,其中均为初等矩阵,构造分块矩阵并作运算: 从式可以看出,可以对进行一系列的初等变换,当块的位置即时,.例1 设,用初等变换求矩阵的逆.解:(I)初等行变换的方

15、法：(II)初等列变换的方法：.(III)同时使用行变换和列变换的方法：.3.2 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,当与都是可逆矩阵时,求证可逆,并求.解法一：因为与都是可逆矩阵,且所以可逆.解法二：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

16、京:高等教育出版社,2003.7.2.姚慕生.高等代数学M.复旦大学出版社,2005.3.3.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),2011,25(9):22-23.4.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,2011,41(19):245-248.5.陈文华.分块矩阵的初等变换及其应用J.大理学院院报,2009,8(8):7.6.严坤妹.分块矩阵的应用J.福建广播电视大学学报,2006,5:71-72.7.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),2007,4:16-17.8.王秀芳.分块矩阵的应用讨论J.

17、连云港师范高等专科学校学报,2008,3:97-98.9.丘维声.高等代数(上)M.北京:清华大学出版社,2010.6,194-211.10.王萼芳,石生明.高等代数M.第三版.北京:高等教育出版社,2003.9(2010重印),181-186.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,

18、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！Many people have the same mixed feelings when planning a trip during Golden Week. W

19、ith heaps of time, the seven-day Chinese请您删除一下内容，O(_)O谢谢！National Day holiday could be the best occasion to enjoy a destination. However, it can also be the easiest way to ruin how you feel about a place and you may become more fatigued after the holiday, due to battling the large crowds. During pea

20、k season, a dream about a place can turn to nightmare without careful planning, especially if you travel with children and older people. As most Chinese people will take the holiday to visit domestic tourist destinations, crowds and busy traffic are inevitable at most places. Also to be expected are

21、 increasing transport and accommodation prices, with the possibility that there will be no rooms available. It is also common that you' ll wait in the line for one hour to get a ticket, and another two hours at the site, to only see a tiny bit of the place due to the crowds. Last year,

22、 428 million tourists traveled in China over the week-long holiday in October. Traveling during this period is a matter that needs thorough preparation. If you are short on time to plan the upcoming "Golden Week" it may not be a bad idea to avoid some of the most crowded places for now. Th

23、ere is always a place so fascinating that everyone yearns for. Arxan is a place like this. The beauty of Arxan is everlasting regardless of the changing of four seasons. Bestowed by nature, its spectacular seasonal landscape and mountains are just beyond word. Arxan is a crucial destination for the

24、recommended travelling route, "China Inner Mongolia Arxan Hailar Manzhouli". It is also the joint of the four prairies across the Sino-Mongolian border, where people gravitate towards the exotic atmosphere mixed with Chinese, Russian, and Mongolia elements. As a historic site for the Yitia

25、n Battle, Arxan still embodies the spirit of Genghis Khan. Walking into Arxan, you will be amazed by a kaleidoscope of gorgeous colors all the year round - the Spring azaleas blooming red in the snow, the Summer sea wavering blue in the breeze, the Autumn leaves painted in yellow covering volcanic t

26、races, and the Winter woods shining white on the vast alpine snowscape. Hinggan League Arxan city is situated in the far eastern area of Inner Mongolia Autonomous Region. Its full name "Haren Arxan" means "hot holy water" in the Mongolian language. Arxan is a tourism city in the

27、northern frontier with a blend of large forest, grand prairies, vast snowfield, heaven lake cluster, thermium, as well as volcanic cluster. It is a rare and unique ecotourism base filled with healthy sunshine, clean air and unspoiled green.  Nestled close to the country's largest virgin for

28、est, and known for its spring and ecological environment, Arxan is marveled at by many tourists as the purest land on earth.  You cannot miss out the Autumn of Arxan. It is definitely the best with brightly-colored scenery full of emotions.  Autumn in the northern part of the country comes

29、 earlier than the South. A September rain followed by the footprints of Autumn brings more colors to the once emerald green mountain and blooming grassland.  Shutterbugs flock to see for themselves the marvel of splendid colors around the mountains and waters, many of whom have travel a long di

30、stance and even camp here only to capture a moment of the nature wonder.  The silver birch turns golden, while the larch is still proudly green. You will find yourself drowned in the intoxicating red of the wild fruits as well as the glamour of flowers in full blown. And your heart will be ling

31、ering on the woods as it's time for the wild fruits to ripe. The picturesque Arxan in Autumn is indeed a fairyland only exists in a dream that satisfies all your fantasies. If it rains heavily on Saturday night, some elderly Chinese will say it is because Zhinu, or the Weaving Maid, is cryi

32、ng on the day she met her husband Niulang, or the Cowherd, on the Milky Way. Most Chinese remember being told this romantic tragedy when they were children on Qixi, or the Seventh Night Festival, which falls on the seventh day of the seventh lunar month, which is usually in early August. This year i

33、t falls on Saturday, August 2. Folklore Story As the story goes, once there was a cowherd, Niulang, who lived with his elder brother and sister-in-law. But she disliked and abused him, and the boy was forced to leave home with only an old cow for company. The cow, however, was a former god who had v

34、iolated imperial rules and was sent to earth in bovine form. One day the cow led Niulang to a lake where fairies took a bath on earth. Among them was Zhinu, the most beautiful fairy and a skilled seamstress. The two fell in love at first sight and were soon married. They had a son and daughter and t

35、heir happy life was held up as an example for hundreds of years in China. Yet in the eyes of the Jade Emperor, the Supreme Deity in Taoism, marriage between a mortal and fairy was strictly forbidden. He ordered the heaven troop to catch Zhinu back. Niulang grew desperate when he discovered Zhinu had

36、 been taken back to heaven. Driven by Niulang's misery, the cow told him to turn its hide into a pair of shoes after it died. The magic shoes whisked Niulang, who carried his two children in baskets strung from a shoulder pole, off on a chase after the empress. The pursuit enraged the empress, w

37、ho took her hairpin and slashed it across the sky creating the Milky Way which separated husband from wife. But all was not lost as magpies, moved by their love and devotion, formed a bridge across the Milky Way to reunite the family. Even the Jade Emperor was touched, and allowed Niulang and Zhinu

38、to meet once a year on the seventh night of the seventh month. This is how Qixi came to be. The festival can be traced back to the Han Dynasty (206 BC-AD 220). Traditionally, people would look up at the sky and find a bright star in the constellation Aquila as well as the star Vega, which are identi

39、fied as Niulang and Zhinu. The two stars shine on opposite sides of the Milky Way. Customs In bygone days, Qixi was not only a special day for lovers, but also for girls. It is also known as the "Begging for Skills Festival" or "Daughters' Festival." In this day, girls will t

40、hrow a sewing needle into a bowl full of water on the night of Qixi as a test of embroidery skills. If the needle floats on top of the water instead of sinking, it proves the girl is a skilled embroiders. Single women also pray for finding a good husband in the future. And the newly married women pr

41、ay to become pregnant quickly. Tradition transformed The love story of Niulang and Zhinu, and the Qixi Festival have been handed down for generations. Yet these ancient traditions and customs are slowly dying out. Many modern Chinese, particularly youngsters, seem to know more about St Valentine'

42、;s Day on February 14, characterized by bouquets of roses, chocolates and romantic candlelight dinners, than they do about their home-grown day for lovers. Even Qixi is nowadays referred to as the "Chinese Valentine's Day." More and more young Chinese people begin to celebrate this day

43、 in a very similar way as that in western countries. Fewer people than ever will gaze at the heavens on Saturday to pick out the two stars shining bright on either side of the Milky Way, that is, if people even know on which day Qixi falls. There are ready reminders dotted about, in the form of big

44、ads saying "Sales on Chinese Valentine's Day!" in shops, hotels and restaurants. But few young women will mark the festival with their boyfriends, or take part in traditional activities to pray for cleverness. A large-scale of exhibition on the Silk Road opened to the public in the Nat

45、ional Museum on Thursday. The exhibition displays more than 400 treasures from Shaanxi province, Gansu province and 14 other regions. It is largest scale exhibition that shows the history of the Silk Road in recent years. As APEC is going on in Beijing, the exhibit aims to show the world t

46、his brilliant section of history. The Silk Road is a series of trade and cultural transmission routes that were central to cultural interaction through regions of the Asian continent. It connected the West and East by linking traders, merchants, pilgrims, monks, soldiers, nomads, and urban dwellers

47、from China and India to the Mediterranean Sea during various periods of time. I met my dear hubby in 2009 the second year I was in my school and it was the first year when he came here. We became colleagues. And there was nothing special happening during that year. He was just a kind and humble brot

48、her graduated from the same college as me. But we did not know each other till he became my colleague and his office desk was just in front of me. There was only a glass between us. So actually we could see each other when we both look up. But it is not the main reason that really draws us close. Ho

49、w we become close to each other was something seems quite natural. We went out to play badminton with other colleagues and we sometimes chatted for a while. He is a man of few words but somehow sometimes will say something really humorous. Then one day, to my great surprise, I found a pot of beautif

50、ul flower on my desk. It was just the flower I had talked about with my colleague the other day, and I thought it was her who put the flower there. But the truth is not. It was the humble guy sitting in front of me. He just simply said ,”well, I caught sight of this flower and thought that its beaut

51、iful and you may like it”. I wonder why I was moved because of these simple words and present. Maybe it is the sincerity and the true self that impress me. I thanked him and some colleagues walked past my desk would appreciate the beautiful flower and they noticed that something romance happened the

52、re. Then later on, there were some little surprises once in a while. Once, there was something wrong with my drawer and I had complained about it unconsciously. And he fixed it for me. Then the other day, he placed several pink diamond roses beside the flower with a piece of paper saying” I love you

53、 more than I can say! Diamond stands for marriage, rose stand for love. I hope you are mine, become my wife. You are the only girl who I really want to spend the rest of my life to stand by.” Although there were some grammar errors but I got what he meant. He meant so clear and so direct.  And

54、actually, I accepted him some days earlier. I was just waiting for him to break the ice. So finally, the common interests and the common understanding brought us together. Some people say the relationship and the feeling will change after getting hitched. You will find that love is no longer strong and there is always lack of passion and whats worse you may find out many bad habits of your partner, which never appear when you are first in love before the marriage. It seems that many faults turn up after you truly living together and see each others face the moment you