几种特殊类型行列式及其计算

毕业论文（设计）作者声明本人郑重声明：所呈交的毕业论文是本人在导师的指导下独立进行研究所取得的研究成果.除了文中特别加以标注引用的内容外，本论文不包含任何其他个人或集体已经发表或撰写的成果作品.本人完全了解有关保障、使用毕业论文的规定，同意学校保留并向有关毕业论文管理机构送交论文的复印件和电子版.同意省级优秀毕业论文评选机构将本毕业论文通过影印、缩印、扫描等方式进行保存、摘编或汇编；同意本论文被编入有关数据库进行检索和查阅.本毕业论文内容不涉及国家机密.论文题目：几种特殊类型行列式及其计算作者单位：数学与信息科学系作者签名：2012年5月31日目录摘要···································································································································································1引言········································································································································································21行列式的定义及性质·······························································································································31.1定义···························································································································································31.2性质···························································································································································32行列式的分类及其计算方法···············································································································42.1箭形（爪形）行列式·····························································································································42.2两三角型行列式······································································································································42.3两条线型行列式······································································································································72.4Hessenberg型行列式····························································································································92.5三对角型行列式····································································································································102.6各行(列)元素和相等的行列式·········································································································112.7相邻两行(列)对应元素相差1的行列式·························································································122.8范德蒙德型行列式································································································································13结束语·································································································································································14参考文献····························································································································································15致谢·········································································································································错误!未定义书签。1几种特殊类型行列式及其计算摘要：行列式的计算是一个普遍的难题.在一些文献中我们已经了解了一些解决它的基本方法，例如:化为上下三角形法，降阶法，加边法，拆项法，递推法，数学归纳法.本文是对几种特殊类型的行列式给以归纳，再根据不同类型给出相应的计算方法.这使得绝大多数行列式能够被归为这其中的某一种，从而能快速简洁的计算出这些行列式.关键词：行列式;爪形;两三角型;两条线型;范德蒙德型SeveralSpecialTypesofDeterminantsandItsCalculationAbstract:Then-thdeterminantcalculationisacommondifficultproblemforstudents.Wehavealreadyknewsomewaysinsomedocumentstosolveit,forexample:themakingdefinition,changingintotriangle(upperandlow),decreasingthedegree,addingthemargin,splittingsomeitems,recursivealgorithmandinduction.Thisarticleaimstoconcludesomespecialkindsofdeterminantsfirstlyandthengivestherelevantcalculationmethods.Thatmademostofthedeterminantscanbeattributedtooneofthatkinds,thenitcanbecalculatedmorequicklyandpithily.KeyWords:Determinant;Claw;“Two-triangle”type;“Two-wire”type;“Vandermonde”type2引言行列式不仅是高等代数的重要内容之一，也是学习其它学科的基础，成为很多学科和领域相当重要的工具，例如在物理学、化学、运筹学等探讨最优化方案时，正是因为成功的应用了行列式来解方程组，才使得问题简单化了，由此可见行列式的计算是一个重要的问题，但同时它也是个比较复杂的问题，特别是高阶行列式，是工程计算中不可或缺的一部分，所以有必要深入研究和归纳高级行列式的计算方法.对这一重要问题，很多文献资料已经做了一些讨论，并给出了相应的结论，如文献[3]讨论了行列式的基本计算方法和技巧，给出了“化零”和“降阶”的基本思想，即先利用行列式的性质做恒等变形化简，使行列式中出现较多零元素，文献[1][10]等具体概括了一些有相同规律的行列式的计算方法，如三线型行列式、两三角型行列式、范德蒙德行列式等.文献[2][9]等通过一些实例的研究，给出了一些重要方法如化三角形法、降阶法、加边法、递推法、数学归纳法等.大部分行列式可以通过变换化为具有某种特点的行列式，进而用相对简便的方法进行计算.本文在上述文献的基础上，首先根据行列式的形态特征对行列式进行分类，总结出几种有某种特点的特殊行列式，再根据不同类型行列式的特点给出相应的计算方法.这样使高阶行列式的计算得到进一步的归纳总结.具有一定的理论意义及应用价值.31行列式的定义及性质1.1定义[3]n级行列式111212122212nnnnnnaaaaaaaaa等于所有取自不同行不同列的个n元素的乘积1212njjnjaaa(1)的代数和,这里12njjj是1,2,,n的一个排列,每一项(1)都按下列规则带有符号:当12njjj是偶排列时,(1)带正号，当12njjj是奇排列时,(1)带有负号.这一定义可写成121212111212122212121nnnnjjjnjjnjjjjnnnnaaaaaaaaaaaa这里12njjj表示对所有n级排列求和.1.2性质[4]性质1.2.1行列互换,行列式的值不变.性质1.2.2某行(列)的公因子可以提到行列式的符号外.性质1.2.3如果某行(列)的所有元素都可以写成两项的和,则该行列式可以写成两行列式的和;这两个行列式的这一行(列)的元素分别为对应的两个加数之一,其余各行(列)与原行列式相同.性质1.2.4两行(列)对应元素相同,行列式的值为零.性质1.2.5两行(列)对应元素成比例,行列式的值为零.性质1.2.6某行(列)的倍数加到另一行(列)对应的元素上,行列式的值不变.性质1.2.7交换两行(列)的位置,行列式的值变号.42行列式的分类及其计算方法2.1箭形（爪形）行列式这类行列式的特征是除了第1行(列)或第n行(列)及主(次)对角线上元素外的其他元素均为零，对这类行列式可以直接利用行列式性质将其化为上(下)三角形行列式来计算.即利用对角元素或次对角元素将一条边消为零.例1计算n阶行列式123231111001000100nnnaaDaaaaa.解将第一列减去第二列的21a倍，第三列的31a倍第n列的1na倍，得122311111000000000nnnaaaaDaa1221nniiiiaaa.2.2两三角型行列式这类行列式的特征是对角线上方的元素都是c,对角线下方的元素都是b的行列式，初看，这一类型似乎并不具普遍性，但很多行列式均是由这类行列式变换而来，对这类行列式，当bc