行列式的計(jì)算畢業(yè)論文--中英文對(duì)照

1、 X X 大 學(xué) 行列式的計(jì)算學(xué)生姓名： 學(xué) 號(hào)： 班 級(jí)： 專 業(yè)：系 別: 指導(dǎo)教師： 行列式的計(jì)算摘 要：行列式是高等代數(shù)研究中的一個(gè)重要工具.本文從行列式的計(jì)算出發(fā)，通過例題，介紹行列式計(jì)算中的一些方法，同時(shí)初步給出了一些特殊行列式的計(jì)算方法，得出了一些關(guān)于行列式計(jì)算的技巧.關(guān)鍵詞：行列式；三角化法；因式定理法；遞推法；數(shù)學(xué)歸納法 引 言行列式出現(xiàn)于線性方程組的求解，它最早是一種速記的表達(dá)式，現(xiàn)在已經(jīng)是數(shù)學(xué)中一種非常有用的工具.行列式是由萊布尼茨和日本數(shù)學(xué)家關(guān)孝和發(fā)明的.同時(shí)代的日本數(shù)學(xué)家關(guān)孝和在其著作解伏題元法中也提出了行列式的概念與算法.1750年，瑞士數(shù)學(xué)家克拉默(1704-17

2、52)在其著作線性代數(shù)分析導(dǎo)引中，對(duì)行列式的定義和展開法則給出了比較完整、明確的闡述，并給出了現(xiàn)在我們所稱的解線性方程組的克拉默法則.稍后，數(shù)學(xué)家貝祖 (1730-1783)將確定行列式每一項(xiàng)符號(hào)的方法進(jìn)行了系統(tǒng)化，利用系數(shù)行列式概念指出了如何判斷一個(gè)齊次線性方程組有非零解.行列式是多門數(shù)學(xué)分支學(xué)科一個(gè)工具，在我們學(xué)習(xí)高等代數(shù)時(shí)，書中只介紹了幾種較簡單的行列式計(jì)算方法，但是在遇到比較復(fù)雜或技巧性比較強(qiáng)的行列式時(shí)，只局限于書上的幾種方法，那解題就有點(diǎn)麻煩.這里我討論了行列式計(jì)算的若干方法，針對(duì)不同的行列式來選擇相對(duì)簡單的計(jì)算方法，來提高解題的效率.1 基本概念的簡單介紹1.1 n級(jí)行列式定義1

3、級(jí)行列式 （1）等于所有取自不同行不同列的個(gè)元素的乘積的代數(shù)和.其中是的一個(gè)排列，的每一項(xiàng)都按下列規(guī)則帶有符號(hào)：當(dāng)是偶排列時(shí)，帶有正號(hào)，當(dāng)是奇排列時(shí)，帶有負(fù)號(hào).1.2 矩陣在敘述行列式的重要公式和結(jié)論以及后面計(jì)算行列式過程中可能要用到矩陣及其有關(guān)概念，所以在這里簡單介紹一下矩陣及其部分概念.定義2 由個(gè)數(shù)排成的行（橫的）列（縱的）的表 （2）稱為一個(gè)矩陣.特別地，當(dāng)時(shí)，（1）稱為（2）的行列式，如果把（2）記作，則（1）表示為.定義3 在行列式中劃去元素所在的第行和第列后，剩下的個(gè)元素按照原來的排法構(gòu)成一個(gè)級(jí)行列式 （3）稱為元素的余子式，記作，而稱為的代數(shù)余子式，記作： （4）定義4 我們把

4、 （5）稱為矩陣（2）轉(zhuǎn)置，記作或，顯然，矩陣的轉(zhuǎn)置是矩陣.定義5 在一個(gè)級(jí)行列式中任意選定行列位于這些行和列的交點(diǎn)上的個(gè)元素按照原來的次序組成一個(gè)級(jí)行列式,稱為行列式的一個(gè)級(jí)子式.2 行列式的性質(zhì)按照行列式的值可分為以下幾類：性質(zhì)1 行列式值為01) 如果行列式有兩行相同，則行列式值為0;2) 如果行列式有兩行成比例，則行列式值為0;3) 行列式中有一行為0，則行列式的值為0.性質(zhì)2 行列式值不變1) 把一行的倍數(shù)加到另一行，行列式值不變, 即 （6）其中.2) 行列互換，行列式值不變, 即= （7）3) 如果行列式的某一行是兩組數(shù)的和，那么它就等于兩個(gè)行列式的和, 這兩個(gè)行列式除這一行外其

5、余與原來行列式對(duì)應(yīng)相同,即 （8）性質(zhì)3 行列式的值改變 一行的公因子可以提出去，或者說用一數(shù)乘以行列式的一行就等于用該數(shù)乘以此行列式 （9）性質(zhì)4 行列式反號(hào)對(duì)換行列式兩行的位置，行列式反號(hào) （10）3 行列式的計(jì)算3.1 一些重要的公式和結(jié)論(1) 行列式按行（或列）展開設(shè)為級(jí)方陣，為的代數(shù)余子式，則 （11） （12）(2) 設(shè)為級(jí)方陣，則 （13）(3) 設(shè)為級(jí)方陣，則 （14）(4) 設(shè)為級(jí)方陣，則 ,但 （15） , （但一般地） （16）(5) (拉普拉斯定理)設(shè)在級(jí)行列式中任意取定了個(gè)行，由這行元素所組成的一切級(jí)子式與它們的代數(shù)余子式的乘積的和等于行列式.(6) 設(shè)為級(jí)方陣，為

6、級(jí)方陣，則：,但是： （17）(7) 范德蒙德行列式 （18）(8) 一些特殊行列式的值 （19）對(duì)角行列式 上三角行列式 下三角行列式 （20） 次對(duì)角行列式 次上三角行列式 次下三角行列式 說明：（19）（20）中的行列式中*號(hào)處的元素不全為零. 3.2 低級(jí)行列式的計(jì)算 3.2.1 利用行列式定義，性質(zhì)例1計(jì)算行列式 解：可以直接按照定義把行列式寫開，得.3.2.2 利用三角化法例2 計(jì)算行列式解：利用三角化法：.3.3 n級(jí)行列式的計(jì)算3.3.1 利用定義3.3.2 逐行（列）相減（加）法3.3.3 利用因式定理法3.3.4 遞推降級(jí)法3.3.5 拆分法3.3.6 數(shù)學(xué)歸納法3.3.7

7、 利用公式和定理參考文獻(xiàn)1 王萼芳，石生明高等代數(shù)M北京大學(xué)數(shù)學(xué)系幾何與代數(shù)教研室前代數(shù)小組編, 1988032 張禾瑞，郝炳新高等代數(shù)M北京高等教育出版社, 1983043 李志慧，李永明高等代數(shù)分析與選講M陜西師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院, 2005094 耿鎖華行列式性質(zhì)的應(yīng)用M南京審計(jì)學(xué)院出版社, 2006015 高麗，郭海清兩類特殊行歷史的計(jì)算M西南民族大學(xué)出版社, 2007066 劉崇華一類行列式的計(jì)算公式M南寧大學(xué)出版社, 200604.7 楊立英，李成群級(jí)行列式的計(jì)算方法與技巧M廣西師范學(xué)院出版社, 200601.8 孫清華，孫昊，李金蘭高等代數(shù)內(nèi)容、方法與技巧M華中科技大學(xué)出版

8、社, 200608.9 毛綱源線性代數(shù)解題方法技巧歸納(第二版)M華中理工大學(xué)出版社, 200706.The calculation of determinantAbstract Determinant is an important tool to study in higher algebra. In this paper, from the determinant calculation by examples, introduces some methods of determinant computation, at the same time, the preliminary ca

9、lculation method is given. Some special determinant, draw some about the determinant calculation skills.Keywords Determinant; triangulation; factorization theorem; recursive method; mathematical inductionIntroductionSolving the determinant in linear equations, it is the first expression is a shortha

10、nd, now is a very useful tool in mathematics. The determinant is invented by Leibniz and the Japanese mathematician Seki takakazu. Contemporary Japanese mathematician Seki Takakazu in his book "V" thematic method solution also proposed the concept and algorithm of determinant.In 1750, the

11、Swiss mathematician Cramer (1704-1752) in his book "linear algebra analysis guide", the definition of the determinant and expansion gives a relatively complete, clear, and gives now we call the solution of linear equations of the Cramer's rule. Later, the mathematician Bei Zu (17 30-17

12、83) will determine the method of determinant each symbol is a systematic concept, using the coefficient determinant points out how to judge a homogeneous linear equations with non-zero solution.The determinant is one branch of mathematics as a tool, we learn in "Higher Algebra", the book d

13、escribes only the determinant of some simple calculation methods, but in the face of the complicated or skills relatively strong determinant, several methods are confined to the book, the problem a bit of trouble. Here I discuss some methods for calculating determinant, the determinant to choose acc

14、ording to different method to calculate the relative simple, to improve the efficiency of problem solving.1 A brief introduction to the Basic Concepts1.1 n determinantDefines 1 levels of determinant （1）Is equal to the algebraic sum of all taken from different lines of different column n elements of

15、the product Where is the 1, 2, n an order, each one of them according to the following rules with symbols: when is even permutation, with positive , when is odd permutation, with a minus sign.1.2 matrixMay be used as matrix and its related concept in the process of the determinant of the determinant

16、 formula and conclusions and back calculation, so here is simple to introduce the concept of matrix and its parts.Definition 2 by the number of into s lines (horizontal) n column (vertical) in Table （2）Known as a matrix.In particular, when, (1) (2) is called the determinant, if (2) denoted as A, the

17、n (1) expressed as a Definition 3In the determinant of In return for element in the i and j columns, the rest of the elements according to the original method consisting of a n-1 determinant of （3）.Known as the cofactor element type, denoted as , while the is called the algebraic type, denoted as: （

18、4）.Definition 4 We call （5）Known as the matrix transpose (2), denoted as or, apparently, transpose of matrix is .Definition 5 In n determinant of D in any of the selected k row k column is located in the intersection of these rows and columns of the elements according to the original order in which

19、a k determinant of M, called a k step determinant of D type.2 Properties of the determinantAccording to the value of determinant can be divided into the following categories:(1) Properties of determinant value is 01) If there are two lines of the same determinant, the determinant value of 0;2) If th

20、e determinant is two in proportion, the determinant value of 0;3) The determinant of a behavior of 0, the determinant of the value of 0(2) Properties of determinant.1) The line ratio to another line, the determinant of invariant, i.e. （6）2) Transpose, determinant value unchanged, i.e.= （7）3) If a ro

21、w determinant is two sets of numbers and, then it is equal to the two determinant and the two determinant, in addition to the line outside the rest with the original determinant corresponding to the same, i.e. （8）(3)Change properties of determinantThe common factor line can be put forward to, or use

22、 a multiplied by the determinant of a is equal to the number is multiplied by this determinant （9）(4)Properties of determinant inverse numberOn line two, the number of determinant （10）3Calculation of determinant3.1 Some important formulas and conclusions(1) The determinant line (or column) expansion

23、Let be n matrix, the cofactor type, then （11） （12）(2) Let A be a n matrix, （13）(3) Let A be a n matrix, （14）(4) Let A, B is n matrix, ,但 （15）, (, but generally ) (16)(5) (Laplasse theorem) In arbitrary n determinant of D in the line, product of algebraic all k type consisted of the k elements and th

24、eir type and is equal to the determinant of D.(6) Let A be a m matrix, B matrix, n, But, （17）(7) Van Redmond determinant (18).(8) Some special determinant value (19) (20).Notes: (19) (20) of the determinant of the elements * are not all zero.3.2Calculation of primary determinant3.2.1 Use of the defi

25、nition of the determinant, properties1 cases of computing determinant of Solution: can be directly according to the definition of the determinant is written, .3.2.2 Uses triangulation method2 cases of computing determinant of Solution: the use of triangulation method 3.3Calculation of level n determ

26、inant3.3.1Using the definition 3.3.2 Row (column) subtract (add) method3.3.3 Factor theorem method3.3.4The recursive degradation method3.3.5 Method3.3.6 Mathematical induction3.3.7Using the formula and theoremReference1 Wang Efang, Ihi Kim algebraic geometry and higher algebra M. Department of Peking University Department of mathematics of algebra group coding,1988.03.2 Zhang Herui, Hao Bingxin. Advanced algebra M. Beijing higher education press, 1983.04.3 Li Z