范德蒙行列式的相關(guān)應(yīng)用外文翻譯

1、精選優(yōu)質(zhì)文檔-傾情為你奉上分類(lèi)號(hào)：_ 學(xué)校代碼：11059 學(xué) 號(hào)：畢業(yè)論文外文翻譯材料 學(xué)生姓名： 學(xué) 號(hào)： 專(zhuān)業(yè)班級(jí)： 數(shù)學(xué)一班 指導(dǎo)教師： 正文：外文資料譯文 附 件：外文資料原文 指導(dǎo)教師評(píng)語(yǔ)： 簽名： 年 月 日范德蒙行列式的相關(guān)應(yīng)用（一）范德蒙行列式在行列式計(jì)算中的應(yīng)用范德蒙行列式的標(biāo)準(zhǔn)規(guī)范形式是：根據(jù)范德蒙行列式的特點(diǎn)，將所給行列式包括一些非范德蒙行列式利用各種方法將其化為范德蒙行列式，然后利用范德蒙行列式的結(jié)果，把它計(jì)算出來(lái)。常見(jiàn)的化法有以下幾種：1.所給行列式各列（或各行）都是某元素的不同次冪，但其冪次數(shù)排列與范德蒙行列式不完全相同，需利用行列式的性質(zhì)（如提取公因式，調(diào)換各行

2、（或各列）的次序，拆項(xiàng)等）將行列式化為范德蒙行列式。例1 計(jì)算解 中各行元素都分別是一個(gè)數(shù)自左至右按遞升順序排列，但不是從0變到。而是由1遞升至。如提取各行的公因數(shù)，則方冪次數(shù)便從0變到. 例2 計(jì)算解 本項(xiàng)中行列式的排列規(guī)律與范德蒙行列式的排列規(guī)律正好相反，為使中各列元素的方冪次數(shù)自上而下遞升排列，將第列依次與上行交換直至第1行，第行依次與上行交換直至第2行第2行依次與上行交換直至第行，于是共經(jīng)過(guò)次行的交換得到階范德蒙行列式： 若的第行（列）由兩個(gè)分行（列）所組成，其中任意相鄰兩行（列）均含相同分行（列）；且中含有由個(gè)分行（列）組成的范德蒙行列式，那么將的第行（列）乘以-1加到第行（列），消

3、除一些分行（列）即可化成范德蒙行列式：例3 計(jì)算解 將的第一行乘以-1加到第二行得：再將上述行列式的第2行乘以-1加到第3行，再在新行列式中的第3行乘以-1加到第4行得：例4 計(jì)算 （1）解 先加邊，那么再把第1行拆成兩項(xiàng)之和，2.加行加列法各行（或列）元素均為某一元素的不同方冪，但都缺少同一方冪的行列式，可用此方法：例5 計(jì)算解 作階行列式：=由所作行列式可知的系數(shù)為，而由上式可知的系數(shù)為：通過(guò)比較系數(shù)得：3.拉普拉斯展開(kāi)法運(yùn)用公式=來(lái)計(jì)算行列式的值：例6 計(jì)算 解 取第1，3，2行，第1，3，列展開(kāi)得：=4.乘積變換法例7 設(shè)，計(jì)算行列式解 例8 計(jì)算行列式解 在此行列式中，每一個(gè)元素都可

4、以利用二項(xiàng)式定理展開(kāi)，從而變成乘積的和。根據(jù)行列式的乘法規(guī)則，其中 對(duì)進(jìn)行例2中的行的變換，就得到范德蒙行列式，于是 = =5.升階法例9 計(jì)算行列式解 將升階為下面的階行列式即插入一行與一列，使是關(guān)于的階范德蒙行列式，此處是變數(shù)，于是故是一個(gè)關(guān)于的次多項(xiàng)式，它可以寫(xiě)成另一方面，將按其第行展開(kāi)，即得比較中關(guān)于的系數(shù)，即得 (二) 范德蒙行列式在多項(xiàng)式理論中的應(yīng)用例1 設(shè)若至少有個(gè)不同的根，則。證明 取為的個(gè)不同的根，則有齊次線形方程組 (2)其中看作未知量因?yàn)榉匠探M（2）的系數(shù)行列式是Vander monde行列式，且所以方程組（2）只有零解，從而有即是零多項(xiàng)式。例2 設(shè)是數(shù)域F中互不相同的數(shù)

5、，是數(shù)域F中任一組給定的不全為零的數(shù)，則存在唯一的數(shù)域F上次數(shù)小于n的多項(xiàng)式，使=，證明 設(shè)由條件，知 （3）因?yàn)榛ゲ幌嗤?，所以方程組（3）的系數(shù)行列式則方程組（3）有唯一解，即唯一的次數(shù)小于的多項(xiàng)式使得，例3設(shè)多項(xiàng)式， ，則不可能有非零且重?cái)?shù)大于的根。證明 反設(shè)是的重?cái)?shù)大于的根，則=0， 進(jìn)而即（4）把（4）看成關(guān)于為未知量的齊次線形方程組則（4）的系數(shù)行列式 = 所以方程組（4）只有零解，從而，所以必有這與矛盾，故沒(méi)有非零且重?cái)?shù)大于的根。附件：(外文資料原文)New proof of the Vander monde determinant and some applications(A)

6、: a new method of proof: mathematical inductionWe on the n for that the inductive method.（1）When ， When the result is right.（2）The Vander monde determinant conclusion assumptions for the class, now look at the level of.in ，Subtracting the rows times, the first rows by subtracting times, that is, a b

7、ottom-up sequentially subtracted from each row on row timeshare有The latter determinant is a Van Dear Mind determinant, according to the induction assumption, it is equal to all possible difference ；Contains difference all appear in front of the consequent conclusion van dear Mend determinant of the

8、level n the establishment of mathematical induction, the proof is completed This result can be abbreviated as even the multiplication signImmediately by the results obtained necessary and sufficient condition for van der Mond determinant is zeroAt least two equal number nThe application of the Vande

9、r monde determinant（一）Vander monde determinant in the determinant calculationThe Vander monde determinant standards form：According to the characteristics of the Vander monde determinant given determinant using various methods, including some non-Vander monde determinant into the Vander monde determi

10、nant, and then use the results of the Vander monde determinant, it calculated. The common method of following：1. Given determinant of the columns (or rows) are different powers of an element, but the number of power arrangement with the Vander monde determinant is not exactly the same, the need to u

11、se the nature of the determinant (such as extraction common divisor, change each line (or column) order, the dissolution of items, etc.) as the determinant of the Vander monde determinant.Example 1.Solutions of elements of each row are a number from left to right in ascending order, but not from 0 t

12、o. But by a delivery rosé. The common factor, such as extraction of each line number of a power from zero change to. Example 2Solution The law of the law of the determinant arranged with the arrangement of the Vander monde determinant on the contrary, to make the columns in the elements of a po

13、wer of frequency from top to bottom in ascending order, the columns uplink switch in turn until the first line, rows sequentially exchanged with the uplink until the second row 2nd row sequentially with uplink switch until the first rows, so after a total ofSub-line exchange Vander monde determinant

14、 of order ： If th row (column) consists of two branches (column), any two adjacent lines (columns) contain the same branch (column); and contains the Vander monde ranks n branches (column)type, then the i-th row (column) multiplied by -1 added to the -row (column) to eliminate some of the branches (

15、column) into the Vander monde determinant：Example3Solution will be the first line of the d multiplied by -1 to the second line we have:Then the second row of the above determinant is multiplied by -1 added to line 3, the new determinant line 3 line 4 was multiplied by -1 added:2 plus line to add law

16、Each row (or column) the different square a power of the whose elements are all the of an element, but are the lack of the the determinant of the same party a power of, available this method:Example 4 Solution for order determinant：= seen by made determinant coefficient, by the coefficients of the a

17、bove equation :By comparing the coefficients obtained:3 Laplace expansion methodsApply the formula = to calculate the value of the determinant：Example 5Dereference lines 1，3，2, 1，3， series expansion：=4 product transformation methodExample 6Set up ，Compute the determinantSolution . Ascending OrderExa

18、mple 7 compute the determinantThe solution l order for the following -order determinantInsert a row with a , order Vander monde determinant, where is the variable, so therefore G H polynomial, it can be written asOn the other hand, the started their first -line, that wasCompare factor that was (B) T

19、he Vander monde determinant polynomial theoryExample 1 set is at least type root, 。Proof of different root, homogeneous linear equations (2)Where as unknown amountVander monde Determinant is because the coefficients of the equations (2), and Equations (2) only the zero solution, thus is zero polynomial.Example 2 Let number different from each other in the number field F, is any number field F Group given not all zero number, then there exists a unique number field