本科畢業(yè)論文淺談分塊矩陣的運算合理性及其在求逆矩陣中的運用

1、分類號 0151.2 論文編號201040432023本 科 生 畢 業(yè) 論 文 淺談分塊矩陣的運算合理性及其在求逆矩陣中的運用姓 名： 院 系： 數(shù)學科學學院 年級專業(yè)：2010級數(shù)學與應用數(shù)學 指導教師： 張裕波 2014年5月目 錄摘 要IIAbstractIII第一章 緒論1第二章 分塊矩陣的概念及其運算合理性12.1 分塊矩陣的概念12.2 分塊矩陣的運算合理性22.2.1 分塊矩陣乘法的合理性22.2.2分塊矩陣的初等變換及其合理性3第三章 分塊矩陣的運算在求逆矩陣中的運用53.1 利用分塊矩陣的思想探討逆矩陣的計算問題53.2 求分塊矩陣的逆矩陣8第四章 結(jié)論9參考文獻10致 謝

2、11摘 要分塊矩陣乘法的合理性是指,在作矩陣乘法時,對矩陣的塊可以像對矩陣的元素一樣對待.分塊矩陣的初等變換與矩陣乘法有十分密切的關系,直接依賴于所涉及到的矩陣乘法是否能進行.分塊矩陣的三類初等變換,其實就是對不分塊的原矩陣進行若干次初等變換.有趣的是,分塊矩陣的初等變換,比普通的初等變換,其“規(guī)?！币蟮枚?是一種“成批量”的初等變換.探討逆矩陣的求法問題,是分塊矩陣運算的典型運用.求逆矩陣的典型例子,反映了分塊矩陣的乘法運算與初等變換之間的深刻聯(lián)系.關鍵詞:分塊矩陣 初等變換 合理性 矩陣求逆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 緒論矩陣是數(shù)學中一個重要的基本概念,是代數(shù)學中重要的研究對象之一,是數(shù)學研究和應用的重要工具,矩陣的性質(zhì)依賴于矩陣中元素的性質(zhì),矩陣由最初的一種工具經(jīng)過近兩個世紀的發(fā)展,形成了一門較為完善的學科矩陣論.矩陣理論應用于許多領域,其中在數(shù)學和工程學中應用最為廣泛.在數(shù)學上,矩陣是求解線性方程組,研究線性空間,線性變換等問題的最要工具,而在一些階數(shù)較高的矩陣中，計算量卻是相當復雜，此時引出分塊矩陣，分塊矩陣也是矩陣論中一個重要的概念,在線性代數(shù)及高等代數(shù)中扮演著不可缺少的角色,分塊矩陣在進行行列式的計算,求解線性方程組,特

7、別是在求逆矩陣的問題上,有非常重要的作用.矩陣分塊,就是將一個高階數(shù)的矩陣看成是由一些低階數(shù)的矩陣組成的,從而將這些低階數(shù)的矩陣看成是高階數(shù)的矩陣的元素,特別是在矩陣的相關運算中,把這些小矩陣看成是數(shù)來處理,為矩陣的運算帶來了許多方便,將矩陣分塊之后,矩陣之間的相互關系,可以看得十分清楚,能夠在解決實際問題中,使問題變得簡單化.本文即是通過查閱相關文獻和學習相關知識后總結(jié)和探究分塊矩陣的運算的合理性以及分塊矩陣在求逆矩陣方面的應用.通過對分塊矩陣的運算合理性及分塊矩陣的初等變換的研究,總結(jié)和探究分塊矩陣在求逆矩陣方面的應用,從而得出分塊矩陣在處理很多問題上帶來方便.第二章 分塊矩陣的概念及其運

8、算合理性2.1 分塊矩陣的概念定義1 在矩陣的行或列之間劃上橫線或豎線將其分為若干部分，稱為對矩陣進行分塊,分了塊的矩陣稱為分塊矩陣.顯然,對于一個的矩陣,最多可以分成塊（一個元素一塊）,最少可以分成一塊（它本身）.注：1、在對矩陣進行分塊的過程中,必須注意分塊的“線”必須穿過整個矩陣且必須是直線，如果在行或列之間所劃直線沒有穿過矩陣，或者所劃的線不是直線而是“折線”，這種分塊分法均是錯誤的.錯誤的分塊方法如： （線條未穿過）2、矩陣的分塊要根據(jù)需要選擇正確的分法，應有利于認識矩陣的結(jié)構(gòu)，便于矩陣的運算和討論有關矩陣的性質(zhì).2.2 分塊矩陣的運算合理性普通矩陣可以進行加（含減）乘（含除和乘方）

9、以及數(shù)乘等運算.因為矩陣的轉(zhuǎn)置、矩陣的初等變換與矩陣的運算有密切聯(lián)系,所以本文也將轉(zhuǎn)置、初等變換視為矩陣的運算,分塊矩陣同樣可以進行加、乘、數(shù)乘、轉(zhuǎn)置及初等變換等運算，并且限于矩陣的運算結(jié)果而言，在對矩陣進行正確分塊的基礎上，通過“塊”運算得出的結(jié)果與不分塊直接運算得出的結(jié)果是一致的.對此,我們僅就分塊矩陣的乘法和初等變換加以說明.2.2.1 分塊矩陣乘法的合理性分塊矩陣乘法的合理性是指,在作矩陣乘法時,對矩陣的塊可以像對矩陣的元素一樣對待.即先對相乘的兩個矩陣作正確的分塊（后一個矩陣的行的分法與前一個矩陣的列的分法一致）,然后把“塊”看成元素,把分塊矩陣的乘法按照普通矩陣的乘法進行（但要嚴格

10、遵守塊的前后順序）,得到的積是由一些“乘積塊”構(gòu)成的新的分塊矩陣,再將每一“乘積塊”按照普通矩陣乘法乘出來,“安放”在相應的位置,從而得到原來兩個矩陣的乘積.這種分了塊再行計算與不分塊直接計算,其結(jié)果是一致的.設，則有:其中；下面說明分塊矩陣乘法的合理性.計算中元.設，其中,，其中,即是設中元素位于中位置.矩陣的第行屬于第個行組,矩陣的第列屬于的第個列組.則有以下式子成立 = .2.2.2分塊矩陣的初等變換及其合理性我們知道,分塊矩陣的初等變換與普通矩陣的初等變換類似,包含三類:(1)互換分塊矩陣的兩塊行或兩塊列；(2)以某個可逆矩陣左乘以分塊矩陣的某一塊行，或右乘以某一塊列；(3)以某個矩陣

11、左乘以分塊矩陣的某一塊行后加到另一塊行上去，或以某個矩陣右乘以分塊矩陣的某一塊列后加到另一塊列上去.（這里假定上面提到的運算都是可以進行的）與普通矩陣一樣,對于分塊矩陣,也有分塊初等矩陣的概念.顯然,分塊矩陣的初等變換與矩陣乘法有十分密切的關系,直接依賴于所涉及到的矩陣乘法是否能進行,即是說,不是任意矩陣（即使可逆）都可以乘到分塊矩陣的某一塊行或塊列上.即使可以乘,左乘與右乘也有嚴格的要求.對于分塊矩陣的初等變換的合理性,我們主要說明:將一個矩陣分了塊以后進行塊的初等變換,其結(jié)果就是對原矩陣進行若干次普通初等變換.對此,設(1)互換的兩塊行（列），顯然相當于多次互換原矩陣的行（列）(2)設可逆

12、，則可以分解成初等矩陣的乘積:，在乘法可以進行的情況下,將左乘以的第一塊行: 容易看出,其結(jié)果相當于對的前行作次初等行變換.（列的情況的說明與此類似）(3)設，我們來考察，通過計算，得： 一般情況的證明與此完全類似,只是符號較多而已,（列變換的證明也與此類似）.由此看來,分塊矩陣的三類初等變換,其實就是對不分塊的原矩陣進行若干次初等變換.第一類分塊初等變換就是互換矩陣的行（列）,第二類分塊初等變換限于對所乘可逆矩陣的那些塊行（塊列）內(nèi)作初等變換,第三類分塊初等變換涉及到其它塊行（塊列）.有趣的是,分塊矩陣的初等變換,比普通的初等變換,其“規(guī)?！币蟮枚?是一種“成批量”的初等變換.因此,分塊初

13、等矩陣、分塊矩陣的初等變換與普通的初等矩陣、初等變換具有類似的性質(zhì).分塊初等矩陣是可逆矩陣,其逆矩陣仍是初等分塊矩陣；矩陣的分塊行（列）初等變換相當于同類分塊初等矩陣左（右）乘以被變換的矩陣；進行分塊初等變換不改變矩陣的秩,如果是方陣,則不改變它的行列式是否為零的屬性（第一類分塊矩陣初等變換不改變行列式的絕對值,第二類分塊矩陣初等變換不改變行列式的值）.第三章 分塊矩陣的運算在求逆矩陣中的運用分塊矩陣的運算在數(shù)學中有廣泛的運用，這里只揭示分塊矩陣的運算在求逆矩陣方面的運用.3.1 利用分塊矩陣的思想探討逆矩陣的計算問題設可逆,則可用初等行變換求的逆,也可用初等列變換求的逆,還可以同時使用行、列

14、初等變換來求.1、行變換的一般方法為：把用初等行變換化為單位矩陣,同時對用相同的行變換即可化得.事實上，設可逆,則有初等矩陣,使得,由此可得, 格式為：用等式表示為: （這里，設,則當式中時,.）2、列變換的一般方法為:把用初等列變換化為單位矩陣,同時對用相同的列變換即可化得.事實上，設可逆,則有初等矩陣,使得,此時,格式為:用等式表示為: （這里，設,則當式中時,.）3、可以把行變換和列變換結(jié)合在一起,同時使用行變換和列變換求.即:取,其中均為初等矩陣,構(gòu)造分塊矩陣并作運算: 從式可以看出,可以對進行一系列的初等變換,當塊的位置即時,.例1 設,用初等變換求矩陣的逆.解:(I)初等行變換的方

15、法：(II)初等列變換的方法：.(III)同時使用行變換和列變換的方法：.3.2 求分塊矩陣的逆矩陣例2 設是一個分塊矩陣,其中為階方陣,為階方陣,當與都是可逆矩陣時,求證可逆,并求.解法一：因為與都是可逆矩陣,且所以可逆.解法二：第四章 結(jié)論分塊矩陣是高等代數(shù)的基本工具,分塊矩陣的運算（包括初等變換）是高等代數(shù)的基本語言.但是分塊矩陣的運算（特別是分塊矩陣的乘法和初等變換）的合理性并不是一目了然.通過本文的討論,我們看到,求逆矩陣的問題集中反映了分塊矩陣的深刻思想內(nèi)涵.分塊矩陣的乘法與初等變換之間有著深刻的內(nèi)在聯(lián)系.參考文獻1.北京大學數(shù)學系幾何與代數(shù)教研室前代數(shù)小組.高等代數(shù)M.第三版.北

16、京:高等教育出版社,2003.7.2.姚慕生.高等代數(shù)學M.復旦大學出版社,2005.3.3.智婕.分塊矩陣的初等變換J.甘肅聯(lián)合大學學報(自然科學版),2011,25(9):22-23.4.張新育,周世國.矩陣的分塊初等變換與分塊初等矩陣及其應用J.數(shù)學的實踐與認識,2011,41(19):245-248.5.陳文華.分塊矩陣的初等變換及其應用J.大理學院院報,2009,8(8):7.6.嚴坤妹.分塊矩陣的應用J.福建廣播電視大學學報,2006,5:71-72.7.高百俊.分塊矩陣的初等變換及應用J.伊犁師范學院學報(自然科學版),2007,4:16-17.8.王秀芳.分塊矩陣的應用討論J.

17、連云港師范高等?？茖W校學報,2008,3:97-98.9.丘維聲.高等代數(shù)(上)M.北京:清華大學出版社,2010.6,194-211.10.王萼芳,石生明.高等代數(shù)M.第三版.北京:高等教育出版社,2003.9(2010重印),181-186.致 謝值此畢業(yè)論文完成之際,特感謝我的論文指導老師張裕波老師,他是一位認真負責,熱情的老師,在整個畢業(yè)論文的撰寫過程中,他都對我進行了認真的指導和詳細的修改,從他的身上我也學到了很多,比如說嚴謹認真、一絲不茍的工作作風.我將永遠記住他曾經(jīng)給予過我的教導，我要向張老師表達我內(nèi)心最崇高的敬意和最衷心的感謝！同時我還要感謝四年來所有任課老師和輔導員及同學們,

18、在我大學四年的學習和生活中,給予了我很多的教導和很大的幫助，是他們的理解、包容、支持使我逐漸成長，從他們身上我學到了很多做人的道理,將使我受益終生.感謝我的家人和好友,他們的關愛和支持永遠是我前進的最大動力,在任何時候,他們都給予我最大的鼓勵和支持,我感謝他們！最后,向?qū)忛單艺撐暮蛥⒓哟疝q的老師們表示衷心的感謝,感謝你們抽出寶貴的時候參加我的論文答辯會,感謝你們對論文不當之處提出的寶貴意見和建議.請您刪除一下內(nèi)容，O(_)O謝謝！Many people have the same mixed feelings when planning a trip during Golden Week. W

19、ith heaps of time, the seven-day Chinese請您刪除一下內(nèi)容，O(_)O謝謝！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