矩阵的初等变换及其应用

在数学中矩阵最早来源于方程组的系数及常数所构成的方阵，现在矩阵是线性代数最基本也是最重要的概念之一。

在线性代数及其许多的问题中都能看到矩阵的身影，它能把抽象的问题用矩阵表示出来，通过对矩阵进行计算得出结果。

作为矩阵的基础及核心，矩阵的初等变换及应用是非常重要的，它能够把各种复杂的矩阵转化成我们需要的矩阵形式，从而使计算变得更加的简便。

本文总结了线性变换在线性代数、初等数论、通信、经济、生物遗传等方面的应用。

关键词：矩阵；初等变换；标准型；逆矩阵；标准型；秩；方程组ABSTRACTMatrix derived from the first phalanx of the coefficients and constants of the equations in mathematics, now matrix is the most fundamental and important concepts of linear algebra, in linear algebra and many other questions can be seen the figure of the matrix, It can abstract the matrix representation, then matrix calculated results. As the foundation and core of the matrix, the elementary transformation matrix and its application is very important, it can conversion a variety of complex matrix into a matrix form we need, then the calculation becomes more simple.This paper summarizes the application of linear algebra, elementary number theory, communications, and economic, biological heredity.Key words:Matrix; Elementary transformation; standard; inverse matrix; standard; rank; equations;1矩阵及其初等变换的概念 (1)2矩阵初等变换的应用 (1)2.1在线性代数中的应用 (2)2.1.1 将矩阵化简为阶梯型和等价标准型 (2)2.1.2矩阵的分块和分块矩阵的初等变换 (3)2.1.3求伴随矩阵和逆矩阵 (4)2.1.4求矩阵的秩，向量组的秩 (5)2.1.5求矩阵的特征值和特征向量 (6)2.1.6 解线性方程组 (7)2.1.7求解矩阵方程 (8)2.1.8化二次型为标准型 (9)2.1.9判断向量组的线性相关性，求其极大线性无关组 (11)2.2在数论中的应用 (11)2.3在通信中的应用 (13)2.4在经济方面的应用 (14)2.5在生物遗传方面的应用 (15)总结 (18)致谢 (19)参考文献 (20)矩阵的初等变换及其应用在线性方程组的讨论中我们看到，线性方程组的一些重要性质反映在它的系数矩阵和增广矩阵的性质上，并且解方程组的过程也表现为对这些矩阵的转化过程，除方程组之外，还有很多方面的问题也都涉及矩阵的概念及其应用，这些问题的研究常常转化为对矩阵的研究，甚至于有些性质完全不同的、表面上完全没有联系的问题，归结成矩阵问题以后却是相同的。

矩阵的初等变换及其应用线性代数第一次讨论课1.导语2.讨论内容目录3.正文4.个人总结导语:矩阵是研究线性代数方程组和其他相关问题的有力工具，也是线性代数的主要研究啊、对象之一。

它的理论和方法在自然科学、工程技术、社会科学等众多领域等都有极其广泛的应用。

矩阵作为一些抽象数学的具体表现，在数学研究中占有极其重要的地位。

本文从矩阵的概念讨论矩阵的运算及性质，进而讨论用途很广的矩阵的初等变换及其应用。

讨论内容目录矩阵的初等变换及其应用1.两个矩阵的等价2.两个矩阵的乘积3.将矩阵化为行阶梯型、行最简形、标准型4.求矩阵的秩5.求可逆矩阵的逆矩阵6.求线性方程组的解7.判断向量组的线性相关性8.求向量组的秩与极大无关组9.求矩阵的对角化矩阵（采用行列初等变换，对角线元素为特征值）10.二次型化为标准形正文一、矩阵的等价1.定义：若矩阵A经过一系列初等行变换化为B矩阵,则称A与B行等价；若矩阵A经过一系列初等列变换化为B矩阵,则称A与B列等价；若矩阵A经过一系列初等变换化为B矩阵,则称A与B等价（相抵）。

2.矩阵的等价变换形式主要有如下几种：1）矩阵的i行（列）与j行（列）的位置互换；2）用一个非零常数k乘矩阵的第i行（列）的每个元；3）将矩阵的第j行（列）的所有元得k倍加到第i行（列）的对应元上去；即如果两个矩阵可通过有限次上述变换中的一个或几个的组合变为一样的，两个矩阵等价。

3.矩阵等价具有下列性质（1）反身性任一矩阵A与自身等价；（2）对称性若A与B等价，则B与A等价；（3）传递性若A与B等价，B与C等价，则A与C等价；注意：矩阵作初等变换是矩阵的一种运算，得到的是一个新矩阵，这个矩阵一般与原矩阵不会相等。

下面举例说明矩阵等价及等价变换：13640824100412204128--?? ?- ? ?-- ?-??13r r +→43213131414331222136413640824100824100412204122041280 412813641364082410082410000300030060000r rr r r r r rr r r r B ++-++-----???? ? ?-- ? ????→???→---- ? ?-------- ? ?→= ? ? ? ?????1231213121310341813601030013001300001000100000000r r r r r r r r r C -------???? ?-- ? ?→→= ?显然，根据矩阵等价的定义，以上变换过程中的每一个矩阵均为等价的，每个步骤都是等价转换。

矩阵初等变换及其应用毕业论文矩阵初等变换及其应用毕业论文摘 要：初等变换是高等代数和线性代数学习过程中非常重要的，使用非常广泛的一种工具。

本文列举了矩阵初等变换的几种应用，包括求矩阵的秩、判断矩阵是否可逆及求逆矩阵、判断线性方程组解的状况、求解线性方程组的一般解及基础解系、证向量的线性相关性及求向量的极大无关组、求向量空间两个基的过渡矩阵、化二次型为标准形。

并用具体例子说明矩阵初等变换在以上几种应用中是如何运用的。

关键词：矩阵 初等变换 初等矩阵在代数的学习过程中，我发现矩阵的初等变换有许多应用，几乎贯穿着始终。

本文将对矩阵的初等变换进行介绍并以具体例子说明矩阵初等变换的七种应用。

虽然这些计算格式有不少类似之处，但是也指出由于这些计算格式有不同的原理，所以它们的应用也有一些明显的区别。

定义1：矩阵的行（列）初等变换是指对一个矩阵施行的下列变换： （1）交换矩阵的两行（列）(交换第i ，j 两行（列），记作()ij ij r c ）；（2）用一个不等于零的数乘矩阵的某一行（列）即用一个不等于零的数乘矩阵的某一行（列）的每一个元素（用数k 乘以第i 行（列），记作()(())i i r k c k ；（3）用某一个数乘矩阵的某一行（列）后加到另一行（列），即用某一数乘矩阵的某一行（列）的每一个元素再加到另一行（列）的对应元素上(第i 行（列）k 倍加到第j 行（列），记作()(())ij ij r k c k 。

初等行、列变换统称为初等变换。

定义2：对单位矩阵I 仅施以一次初等变换后得到的矩阵称为相应的初等矩阵，分别记为第1、2、3类行（列）初等矩阵为()ij ij R C ，()(())i i R k C k ，()(())ij ij R k C k ，有ij R =ij C =10111⎛⎫ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪ ⎪⎝⎭ ()i R k =()i C k =1k1⎛⎫ ⎪⎪⎪ ⎪ ⎪ ⎪⎝⎭()ij R k =()ij C k =11j 11i k⎛⎫ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪ ⎪⎝⎭行行 初等变换与初等矩阵之间有下列基本性质。

㊀㊀㊀㊀㊀㊀矩阵的初等变换及其应用矩阵的初等变换及其应用Һ顾江永㊀（宿迁学院文理学院，江苏㊀宿迁㊀２２３８００）㊀㊀ʌ摘要ɔ矩阵的初等变换在代数学中具有重要的地位，本文给出了运用初等变换求解方程组的基础解系㊁特征值㊁多项式的最大公因式和Ｊｏｒｄａｎ标准形相似变换矩阵等方法，这些方法具有直观㊁简捷㊁有效等特点．ʌ关键词ɔ初等变换；基础解系；最大公因式；相似变换矩阵ʌ基金项目ɔ２０１９江苏省高校教学研究一般项目（２０１９ＳＪＡ１９９７）一㊁引㊀言矩阵的初等变换包括矩阵的初等行变换和矩阵的初等列变换，矩阵的初等行（列）变换有三种形式［１］：（１）交换两行（列）；（２）任一行（列）的ｋ倍（ｋʂ０）；（３）任一行（列）的ｋ倍加到另一行（列）．在代数学中，矩阵的初等变换有着非常重要且广泛的应用，它常被应用于行列式的计算㊁方程组以及矩阵方程的求解㊁向量线性关系的判定㊁求矩阵的秩以及逆㊁λ－矩阵的不变因子和矩阵的Ｊｏｒｄａｎ标准形等．张家宝给出了初等变换求逆的几种方法［２］；石擎天等研究了初等变换求解方程组的特殊方法［３］；于莉琦等介绍了初等变换在行列式㊁矩阵和方程组中的应用［４］．本文给出了矩阵的初等变换求解方程组的基础解系㊁最大公因式和Ｊｏｒｄａｎ标准形的相似变换矩阵等方法及应用．二㊁预备知识引理１［５］㊀设矩阵Ａｍˑｎ的秩为ｒ，且Ａｍˑｎ＝ＰＥｒ０００æèçöø÷Ｑ，其中Ｐｍˑｍ，Ｑｎˑｎ为可逆矩阵，则有Ｐ－１００Ｅｎæèçöø÷ＡＥｎæèçöø÷Ｑ－１＝Ｅｒ０００Ｑ－１æèççöø÷÷．证明㊀因为Ａｍˑｎ＝ＰＥｒ０００æèçöø÷Ｑ，所以Ｅｒ０００æèçöø÷＝Ｐ－１ＡｍˑｎＱ－１，故Ｐ－１００Ｅｎæèçöø÷ＡＥｎæèçöø÷Ｑ－１＝Ｐ－１ＡＥｎæèçöø÷Ｑ－１＝Ｐ－１ＡＱ－１Ｑ－１æèçöø÷＝Ｅｒ０００Ｑ－１æèççöø÷÷，注：引理１给出了化一个矩阵为标准形的求Ｑ－１的方法．引理２㊀设矩阵Ａｍˑｎ的秩为ｒ，则矩阵ＡＥｎæèçöø÷仅经初等列变换可以化为β１，β２， ，βｒ，０， ，０Ｑ－１æèçöø÷，其中β１，β２， ，βｒ线性无关，且ＡＱ＝β１，β２， ，βｒ，０， ，０()．证明㊀因为Ａｍˑｎ的秩为ｒ，所以Ａｍˑｎ的列秩等于ｒ，即矩阵Ａｍˑｎ列向量组的最大线性无关组由ｒ个向量构成，不妨设为β１，β２， ，βｒ，故由初等变换的性质可得ＡＥｎæèçöø÷仅经初等列变换可以化为β１，β２， ，βｒ，０， ，０Ｑ－１æèçöø÷．引理３［６］㊀设Ａ是数域Ｐ上的ｎ阶方阵，将矩阵λＥ－Ａ经初等变换化为上三角形矩阵ｆ１（λ）０ ０∗ｆ２（λ）０︙︙⋱︙∗∗ｆｎ（λ）æèççççöø÷÷÷÷，则ｆｉ（λ）＝０（ｉ＝１，２， ，ｎ）在数域Ｐ上的根即为矩阵Ａ的全部特征根．证明㊀根据初等变换的性质可知，初等变换不改变λＥ－Ａ＝０的根，故ｆ１（λ）０ ０∗ｆ２（λ） ０︙︙⋱︙∗∗ｆｎ（λ）＝ｆ１（λ）ｆ２（λ） ｆｎ（λ）＝０的根即为矩阵Ａ的全部特征根．引理４㊀设ｆ１（ｘ），ｆ２（ｘ）， ，ｆｓ（ｘ）是数域Ｐ上的多项式，且ｆ１（ｘ），ｆ２（ｘ）， ，ｆｓ（ｘ）()Ｔ经初等行变换化为ｄ（ｘ），０， ，０()Ｔ，则ｄ（ｘ）即为ｆ１（ｘ），ｆ２（ｘ）， ，ｆｓ（ｘ）的最大公因式．证明㊀由辗转相除法原理直接可得［１］．三㊁主要结论定理１㊀设齐次线性方程组Ａｍˑｎｘ＝０，其系数矩阵Ａｍˑｎ的秩为ｒ，且Ａｍˑｎ＝ＰＥｒ０００æèçöø÷Ｑ，又设Ｑ－１＝（η１， ，ηｒ，ηｒ＋１， ，ηｎ），则ηｒ＋１，ηｒ＋２， ，ηｎ是线性方程组Ａｍˑｎｘ＝０的基础解系．证明㊀设Ｑｘ＝ｙ１︙ｙｒ︙ｙｎæèçççççöø÷÷÷÷÷＝ＹｒＹｎ－ｒæèçöø÷，由Ａｍˑｎｘ＝ＰＥｒ０００æèçöø÷Ｑｘ＝ＰＥｒ０００æèçöø÷ＹｒＹｎ－ｒæèçöø÷＝０，可得Ｙｒ＝ｙ１︙ｙｒæèççöø÷÷＝０，所以ｘ＝Ｑ－１ＹｒＹｎ－ｒæèçöø÷＝Ｑ－１０︙０ｙｒ＋１︙ｙｎæèççççççöø÷÷÷÷÷÷．㊀㊀㊀㊀㊀令Ｑ－１＝（η１， ，ηｒ，ηｒ＋１， ，ηｎ），则ｘ＝ｙｒ＋１ηｒ＋１＋ｙｒ＋２ηｒ＋２＋ ＋ｙｎηｎ．因为Ｑ是可逆矩阵，则ηｒ＋１，ηｒ＋２， ，ηｎ线性无关，所以ηｒ＋１，ηｒ＋２， ，ηｎ为方程组的一个基础解系．定理２［７］㊀设Ａ是数域Ｐ上的ｎ阶方阵，矩阵λＥｎ－ＡＥｎæèçöø÷经初等变换化为φ１（λ）０⋱０φｎ（λ）Ｑ（λ）æèççççöø÷÷÷÷（其中初等行变换只能在前ｎ行进行）．设Ｑ（λ）的第ｊ列为ｑｊ（λ），若λ－λ０()ｋ为φｊ（λ）的初等因子，则Ａｑｊ（λ０），ｑᶄｊ（λ０）１！，ｑᵡｊ（λ０）２！， ，ｑ（ｋ－１）ｊ（λ０）（ｋ－１）！æèçöø÷＝ｑｊ（λ０），ｑᶄｊ（λ０）１！，ｑᵡｊ（λ０）２！， ，ｑ（ｋ－１）ｊ（λ０）（ｋ－１）！æèçöø÷λ０１００λ００︙︙⋱１００λ０æèççççöø÷÷÷÷．证明㊀由题设知，存在可逆矩阵Ｐ（λ），Ｑ（λ），使得Ｐ（λ）λＥｎ－Ａ()Ｑ（λ）＝φ１（λ）０⋱０φｎ（λ）æèççöø÷÷．因为ｑｊ（λ）是Ｑ（λ）的第ｊ列，所以Ｐ（λ）λＥｎ－Ａ()ｑｊ（λ）＝（０， ，０，φｊ（λ），０， ，０）Ｔ．又设ｑｊ（λ）的幂级数展开式为ｑｊ（λ）＝ｑｊ（λ０）＋ｑᶄｊ（λ０）１！λ－λ０()＋ｑᵡｊ（λ０）２！λ－λ０()２＋ ，代入Ｐ（λ）λＥｎ－Ａ()ｑｊ（λ）＝（０， ，０，φｊ（λ），０， ，０）Ｔ，得λ０Ｅｎ－Ａ()ｑｊ（λ０）＝０，λ０Ｅｎ－Ａ()ｑᶄｊ（λ０）＋ｑｊ（λ）＝０，λ０Ｅｎ－Ａ()ｑ（ｋ－１）ｊ（λ０）（ｋ－１）！＋ｑｋ－２()ｊ（λ０）ｋ－２()！＝０．上面等式两边相加㊁移项并提取矩阵Ａ可得Ａ(ｑｊ（λ０），ｑᶄｊ（λ０）１！，ｑᵡｊ（λ０）２！， ，ｑ（ｋ－１）ｊ（λ０）（ｋ－１）！)＝(ｑｊ（λ０），ｑᶄｊ（λ０）１！，ｑᵡｊ（λ０）２！， ，ｑ（ｋ－１）ｊ（λ０）（ｋ－１）！)λ０１００λ０ ０︙︙⋱１００λ０æèççççöø÷÷÷÷．四㊁应用举例例１㊀求多项式ｆ１（ｘ），ｆ２（ｘ），ｆ３（ｘ）的最大公因式，其中ｆ１（ｘ）＝ｘ４＋２ｘ３＋４ｘ２＋３ｘ＋２，ｆ２（ｘ）＝ｘ４＋ｘ３＋３ｘ２＋ｘ＋２，ｆ３（ｘ）＝ｘ３＋２ｘ２＋３ｘ＋２．解㊀因为ｆ１（ｘ）ｆ２（ｘ）ｆ３（ｘ）æèççöø÷÷＝ｆ１（ｘ）－ｆ２（ｘ）ｆ２（ｘ）－ｘｆ３（ｘ）ｆ３（ｘ）æèççöø÷÷＝ｘ３＋ｘ２＋２ｘ－ｘ３－ｘ＋２ｘ３＋２ｘ２＋３ｘ＋２æèççöø÷÷＝ｘ３＋ｘ２＋２ｘｘ２＋ｘ＋２ｘ２＋ｘ＋２æèççöø÷÷＝ｘ３＋ｘ２＋２ｘｘ２＋ｘ＋２０æèççöø÷÷＝ｘ２＋ｘ＋２００æèççöø÷÷，所以由引理４知，ｆ１（ｘ），ｆ２（ｘ），ｆ３（ｘ）的最大公因式为ｄ（ｘ）＝ｘ２＋ｘ＋２．例２㊀求齐次线性方程组ｘ１＋ｘ２＋ｘ３＋ｘ４＋ｘ５＝０，３ｘ１＋２ｘ２＋ｘ３＋ｘ４－３ｘ５＝０，５ｘ１＋４ｘ２＋３ｘ３＋３ｘ４－ｘ５＝０{的基础解系．解㊀对系数矩阵Ａ施行初等行变换如下Ａ＝１１１１１３２１１－３５４３３－１æèççöø÷÷ ｒ２－３ｒ１ｒ３－５ｒ１１１１１１０－１－２－２－６０－１－２－２－６æèççöø÷÷ ｒ１＋ｒ２ｒ２ˑ（－１）ｒ３－ｒ２１０－１－１－５０１２２６０００００æèççöø÷÷．又１０－１－１－５０１２２６１０００００１０００００１０００００１０００００１æèçççççççöø÷÷÷÷÷÷÷ ｃ３＋ｃ１ｃ４＋ｃ１ｃ５＋５ｃ１１０００００１２２６１０１１５０１０００００１０００００１０００００１æèçççççççöø÷÷÷÷÷÷÷ ｃ３－２ｃ２ｃ４－２ｃ２ｃ５－６ｃ２１０００００１０００１０１１５０１－２－２－６００１０００００１０００００１æèçççççççöø÷÷÷÷÷÷÷则由引理２知，方程组的基础解系为η１＝（１，－２，１，０，０）Ｔ，η２＝（１，－２，０，１，０）Ｔ，η３＝（５，－６，０，０，１）Ｔ．ʌ参考文献ɔ［１］王萼芳，石生明．高等代数（第五版）［Ｍ］．北京：高等教育出版社，２０１９：５．［２］张家宝．浅谈求逆矩阵的几种方法［Ｊ］．数学学习与研究，２０２０（１０）：４－５．［３］石擎天，黄坤阳．线性方程组求解及应用［Ｊ］．教育教学论坛，２０２０（１２）：３２５－３２７．［４］于莉琦，高恒嵩．初等变换概述［Ｊ］．数学学习与研究，２０１９（０６）：１１６．［５］徐仲，陆全，等．高等代数考研教案（第２版）［Ｍ］．西安：西北工业大学出版社，２００９．［６］卢博，田双亮，等．高等代数思想方法及应用［Ｍ］．北京：科学出版社，２０１７．［７］朱广化．关于‘相似变换矩阵的简单求法“的改进［Ｊ］．数学通报，１９９４（１１）：４４－４６．。

矩阵初等变换：从入门到实践
矩阵初等变换是线性代数重要的基础知识，也是机器学习和人工
智能领域必须掌握的技能。

本文将从基本概念到实际应用，全面深入
地介绍矩阵初等变换的相关知识。

什么是矩阵初等变换？矩阵初等变换指的是对矩阵的行、列进行
一些基本的变换操作，比如交换矩阵的某两行（列）、将某一行（列）中的元素乘以一个非零常数、将某一行（列）加上另一行（列）的k
倍等。

通过矩阵初等变换，我们可以改变矩阵的性质，比如行列式、秩，同时也可以解决某些线性方程组的求解问题。

矩阵初等变换有哪些基本形式？根据变换的形式，矩阵初等变换
可以分为三类：交换两行（列）、将某一行（列）中的元素乘以一个数、将某一行（列）加上另一行（列）的k倍。

需要注意的是，矩阵
初等变换对应的变换矩阵是方阵，也就是说，如果我们进行一次矩阵
初等变换，那么原矩阵的行列式和秩都不会改变。

矩阵初等变换的应用有哪些？矩阵初等变换在线性代数和数学计
算中有着广泛的应用。

我们可以通过矩阵初等变换解决线性方程组的
求解问题，可以判断矩阵的线性相关性，可以求取矩阵的逆矩阵，还
可以将高斯-约旦消元法的过程表示成矩阵初等变换的形式，方便进行
计算。

在机器学习中，矩阵初等变换也有着重要的应用。

比如，我们
可以通过初等变换将数据标准化为均值为零、方差为一的正态分布，
也可以进行特征值分解和奇异值分解等，从而进行降维和信息提取。

总结：矩阵初等变换是线性代数中的重要内容，在数学计算和机器学习领域都有着广泛的应用。

我们应该深入了解矩阵初等变换的各种形式和应用，从而更好地掌握线性代数和机器学习的相关知识。

摘要本文探讨矩阵初等变换的性质及其在代数中的若干应用，主要从矩阵的逆、矩阵的秩、求解线性方程组及矩阵方程、求一元多项式的最大公因式、求解指派问题等若干方面进行阐述。

关键词：矩阵的初等变换；矩阵的秩；可逆矩阵；线性方程组；最大公因式AbstractThis paper is mainly to discuss the application of the elementary transfor mation of matrix in algebra, using matrix elementary transformation to solve th e matrix inverse, matrix rank, solving linear equations and matrix equations, on e yuan polynomial greatest common divisor, solving assignment problem of the se aspects of the application.Keywords:Elementary transformation of matrix;Matrix rank;Invertible matrix;System of linear equations;Greatest common factor目录1 引言 ............................. 错误!未定义书签。

2 矩阵的初等变换及其性质 (1)2.1 矩阵初等变换的定义.......................... 错误!未定义书签。

2.2 矩阵初等变换相关性质 (2)3 矩阵初等变换的若干应用 (2)3.1 利用矩阵初等变换求矩阵的逆 (1)3.2 利用矩阵的初等变换来求矩阵的秩 (5)3.3 利用矩阵初等变换求解线性方程组及矩阵方程 (7)3.4 利用矩阵的初等变换求一元多项式最大公因式 (11)3.5 利用矩阵初等变换解决指派问题 (13)参考文献 (16)矩阵初等变换的性质及其应用矩阵及其理论在众多领域中都发挥着重要的作用,而矩阵的初等变换是矩阵理论的核心和灵魂。

矩阵的初等变换及其应用
矩阵初等变换包括三种变换：
1.交换任意两行或列。

2.用一个数与一行或列中所有数相乘。

3.把某一行或列中的某个数加上另一行或列中对应的数的k倍。

初等变换的主要应用有：
1.解线性方程组：通过初等变换把系数矩阵化为一个容易求解的三角矩阵。

2.计算矩阵的秩：通过初等变换把矩阵化为阶梯形矩阵，可以方便地求出矩阵的秩。

3.求逆矩阵：通过初等变换把原矩阵化为一个对角矩阵，然后对角线上的元素取倒数得到逆矩阵。

矩阵的初等变换及其应用（Elementary transformation of matrixand its application）Elementary transformation of matrix and its applicationWang DanElementary transformation of matrix and its applicationAbstractElementary transformation of matrix is an important method of studying matrix, and it is the core of application in linear algebra. This paper introduces some concepts and properties associated with the matrix, on the basis of matrix rank, the basis for judgment matrix is invertible, after inverse matrix equations, eigenvalues and eigenvectors, two types of standard form, and illustrate the application of elementary transformation of matrix in the above is how to play the role of.Keywords: matrix, elementary transformation, applicationThe, elementary, transformation, of, matrix, and, its, applicationsAbstractElementary transformation matrix is an important means of Matrix is the core linear algebra applications. This article briefly describes some of the concepts and propertiesassociated with the matrix as a basis, the rank of a matrix to determine whether a matrix is reversible after inverse matrix, seeking basic solutions line equations find eigenvalues, and eigenvectors, quadratic standard Shape and so on. Illustrate the elementary transformation matrix in the above applications is how to play a role.Keywords:, matrix, elementary, transformation, applicationCatalog1. introduction 62. the related concepts of matrix 72.1 definition of matrix 72.2 transpose of matrix 72.3 elementary transformation of matrix and elementary matrix 73. the application of elementary transformation of matrix 83.1, the rank of the matrix 83.2 the inverse matrix of the matrix 103.3 using elementary transformation to solve matrix equation 113.4 find the solution of linear equations 12The conditions for the existence of nonzero solutions of 3.4.1 homogeneous linear equations are 13Conditions for the existence of solutions of 3.4.2 nonhomogeneous linear equations 143.5 find the eigenvalues and eigenvectors of the matrix 153.6, use elementary transformation, two times as standard type 17Summary 19References 191. introductionIn the course of studying linear algebra, I find that the elementary transformation of matrix is very extensive and runs through the whole chapter. It is the key to solve the problem in linear algebra. Linear equations is the beginning of the elementary transformation matrix, the matrix effect can also be said to be of linear algebra, each knowledge point of linear algebra and linear algebra and matrix are closely related, each in mathematics both can play a role. Biology, economics, physics, cryptography requires knowledge of mathematics, the significance of matrix elementary transformation of matrix, as can be imagined, is the complex matrix into a simple form is easy to calculate and understand.In real life, many aspects involve the knowledge of matrices,In studying the virtual aircraft model, we will find that the operation of the matrix plays a crucial role. The plane surface appears to be smooth, but the geometric structure is perplexing, the flow equation is more difficult, must also consider other external factors, but we use the matrix knowledge to be able to solve the problem very well. There are many other applications, for example, matrix eigenvalues and eigenvectors is the key to solve many problems in physics, mechanics and engineering technology; now the game company and Bank Account confidential security, but also the use of matrix theory invented the matrix card; simulation in equipment monitoring system in engineering, radio and television; large screen display works, TV teaching, command and control center etc. mainly used matrix switcher and so on.2. concepts related to matrices2.1 definition of matrixTable is a rectangular matrix. Similar to the cross and the determinant is called a row, called vertical columns, with a line, the line and the line and the row element matrix for short note.Transpose of the 2.2 matrixLet a matrix be called a matrixFor the transpose of the matrix, rememberElementary transformation and elementary matrix of 2.3 matrices1, the following three transformations called matrices, called the matrix of the primary row (column) transform, collectively referred to as the elementary transformation of the matrix:(1) the two row (column) of the exchange matrix(2) the elements of a row (column) of a matrix are multiplied by a nonzero constant(3) a constant of the elements of a row (column) of a matrix added to the corresponding element of another row (column)Elementary row and column transformations are collectively referred to as elementary transformations2. The matrix obtained by elementary transformation of a unit matrix is called elementary matrix.Three types of elementary matrices:(1) elementary commutative matrices: the second and the second lines of the commutative unit matrix(2) the elementary multiplied matrix: the row (column) of the unit matrix takes the nonzero constant, i.e.(3) elementary doubly matrix: the first row of a unit matrix is added to the first line, or the first row is multiplied to the next columnIf the matrix is transformed into a matrix by a finite elementary transformation, it is said to be equivalent3, matrix equivalence has the following properties:(1) reflexivity, that is, the self equivalence of any matrix;(2) symmetry, that is, the equivalence of any matrix, if and equivalence;(3) transitivity is equivalent to any matrix, and if and equivalence, equivalence, and equivalence;The application of elementary transformation of 3. matrices3.1, the rank of the matrixMany methods for matrix rank, general definition method, elementary transformation method, formula method and comprehensive method, but when the specific element of the matrix is known, using elementary transformation method is for non zero row (column) number.The highest order of a nonzero divisor defined in a 3.1.1 matrix is called the rank of a matrix. That is, there is a rank order of no 0, and all orders of variables (if any) are 0, then the rank of the matrix is (or / or rank)(1)(2) the rank of the zero matrix is 0(3) the rank of a ladder matrix = the number of nonzero rows in a rowTheorem 3.1.1 the elementary transformation of a matrix does not change the rank of a matrixTheorem 3.1.2 row rank of a matrix = row rank of a matrixTheorem 3.1.3, the equivalent matrices have the same rank, but their inverse is not true, that is, the matrices with the same rank may not be equivalent, and the matrices of the same type and the same rank are equivalent to each otherFind the rank of a matrix, and give a brief introduction of the most common method:(1) definition method:If the matrix has a nonzero order, and all the sub orders (if any) are all 0, then.If there is a nonzero order in the matrix, and all of the order variables containing this order are 0.The usage of matrix rank can be calculated with simple formula omit a lot.(2) the number of zero rows in Central Africa is the rank of the matrix.This is because the elementary row transformation does not change the rank of the matrix, in addition, it can be transformed into a column ladder rank by the elementary column transformation, and the elementary transformation can be used as the standard form to obtain the rank.Example 1 find the rank of a matrix.Solution 1: take the 2 order of the upper left of the matrixHowever, there are only 3 lines in the matrix, so it is necessary to find the 3 order of the variables contained in the matrix.Solution 2: to do elementary row transformationDue to non-zero behavior 2.It can be seen that the definition method is only suitable for the calculation of simple matrix, but if it is a higher order matrix, it is very inconvenient to calculate.3.2, the inverse matrix of the matrixThe definition of 3.2.1 is set as a square matrix, if the order matrix existsHere is the rank unit matrix, which is called the invertible matrix, and is called the inverse matrix.Note (1) if it is invertible, its inverse matrix is unique, and the inverse matrix is;(2) the invertible problem of the matrix is the case of the opponent's matrix.Set the invertible matrix of order, and the inverse matrix is as follows:Example 2 is set up as a invertible square matrix, and the resulting matrix is denoted by the following line and column(1) proved to be reversible;(2) seekingProof: (1) since the left multiplication of the elementary matrix corresponds to the two rows of the interchange, so there isBecause, so the matrix is reversible(2)Example 3 uses the elementary transformation of the matrix to find the inverse matrix of the matrixSolution:soIn short, we in the inverse matrix with elementary transformation, we must first selected by elementary row transformation or elementary column transformation, note that if using elementary row transformation must be from first to last by elementary row transformation, using elementary column transformation must be from first to last by elementary column transformation.But in the inverse does not need to check whether the reversible matrix, elementary transformation can be directly obtained, if the simplest form of a square matrix transform unit is not left after the show, the original matrix is irreversible.3.3 using elementary transformation to solve matrix equation(1) if it is reversible, then(2) if it is reversible, then(3) if both are reversible, thenFirst of allAgainThis can be obtainedThe matrix equations of type can only be elementary row transformations (on the left); the pair can only be elementary column transformations (on the right)Example 4 solving matrix equationSolution: let the original equation be...therefore3.4 solving the system of linear equationsSet a system of linear equations with unknown quantitiesIts matrix form is,Among them,,,The coefficient matrix called linear equation is called the augmented matrix.Conditions for nonzero solutions of 3.4.1 homogeneous linear equations(1) the necessary and sufficient condition for the existence of nonzero solutions of homogeneous linear equations is the rank of the coefficient matrix(2) when the number of equations of a homogeneous linear equation group is less than the number of unknown quantities (m<n), there must be nonzero solutions(3) if the order matrix is square, the system of equations has nonzero solution(4) if the order matrix is square, then the system of equations has only zero solutionFirst, the coefficient matrix is transformed into a ladder matrix by using elementary row transformation, and if there is only zero solution, if there is a nonzero solution, it continues to be calculated;The ladder? Matrix to the simplest form, a non zero row non zero element corresponding to the unknown quantity, the unknown amount of free unknown quantity, revenuer, after making one of a free variable is 1, the remaining 0, basic system of solutions can be obtained.The linear combination of the solutions of the parameters is the general solution of the equationExample 5 solving linear equationsSolution: the coefficient matrix is transformed into the simplest form by elementary row transformationsoThat is, there are 2 free unknownsWith the same set of equationsFor the selection of free unknown, and transferred toThe general solution is()Represented as a vector matrixConditions for the existence of solutions of 3.4.2 nonhomogeneous linear equations(1) if the set is a matrix, then the necessary and sufficient condition for the solution of the nonhomogeneous linear equation set is that the rank of the coefficient matrix is equal to the rank of the augmented matrix(2) if the set of nonhomogeneous linear equations is solvable, thenThe solution is unique and the second set of equations has only zero solutions.(3) there are infinitely many solutions to the system of nonhomogeneous linear equations(4) the solution of a system of nonhomogeneous linear equations without elementExample 6 for solving nonhomogeneous linear equationsSolution: an elementary row transformation of the augmented matrixThat wasTherefore, the general solution of the original equation set is any constant3.5 find the eigenvalues and eigenvectors of the matrixThe definition of 3.5.1 is a matrix of order, if there exists a number and a zero dimensional column vector, theThat isSet up is called an eigenvalue of a square matrix, and nonzero column vectors are called eigenvectors of the square corresponding to (or belong to) eigenvaluesThe characteristic polynomial of a 3.5.2 determinant (or) called a matrix (Note: the sub polynomial of a characteristic polynomial is) is a characteristic equation of a matrix:Let the order matrix be the unit matrix of the order, the eigenvalues of the matrix, and the matrixWith the elementary transformation, the upper triangular matrix can be obtained, and the product of the elements on the principal diagonal of the matrix is 0The value is the eigenvalue of the matrix.Example 7 uses the elementary transformation method of matrix to find the eigenvalues and eigenvectors of the matrixSolution:The product of the principal diagonal elements of the order is zero, i.e.EigenvalueThenTherefore, the corresponding eigenvectors areAll the corresponding eigenvectors are.WhenTherefore, the corresponding eigenvectors areThe entire feature vector at this time is.3.6, use the elementary transformation, and the two form is the standard typeTwo order homogeneous polynomials with variablesReferred to as the "yuan two times", referred to as the "twotimes".Order, rememberThen the two type can be expressed asA matrix of symmetric matrices of two order.When a series of elementary column transformations are applied to a matrix, the same elementary row transformation is applied to the block,When the block diagonal matrixWhen the child blocks are reduced, the. At this point, if the order, then into a standard shapeExamples are 8 and two times as standard.Solution: the quadratic matrix is twoImplementing elementary transformationIn this way, by coordinate transformation, of whichThe two form is a standard shapeNote: two types can be standardized in a variety of ways, and their standard shapes are not unique.Sum upTo solve some problems in algebra when using the elementary matrix transform can simplify the problem, such as the two type as the standard type, in addition to using elementary transformation method, also can be calculated using the orthogonal transformation method and collocation method, comparison of elementary transformation is simple, easy to calculate, easy to understand. The elementary transformation matrix has many applications in solving computational problems of linear algebra, the calculation format has many similar places, once mastered the operation of the matrix, we analyze and solve the equations of the ability will be greatly enhanced.In a word, the elementary transformation of matrix is an important method of calculation in linear algebra. We can use matrix elementary transformation to compute the rank of matrix, inverse matrix and matrix equation. With the development of science and technology, matrix has been applied to the natural, social, engineering, economic and other fields, and artificial intelligence, mobile phone communication and algorithm design and general analysis, the matrix in its application is communication optimization. We can not confine ourselves to the study of books. We should integrate theory with practice and make better use of theoretical knowledge to solve practical problems.Reference[1] 、 pre algebra group, Department of geometry and algebra, Department of mathematics, Peking University. Advanced Algebra (Third Edition), higher education press,.2003[2] Ma Juxia, Wu Yuntian. Linear Algebra (Second Edition). National Defense Industry Press.2009.8[3] Chen Zhizhong. Refined refining of linear algebra. Beijing Normal University press,.2006[4], Li Zhihui, Li Yongming. Typical problems and methods in advanced algebra. Science Press,.2008[5], Kang Yonghai, Zhu Baoyan. Application of elementary transformation of matrix in solving problems [J]. Journal of Songliao University (NATURAL SCIENCE EDITION),.1998 (3)[6], Yao Gang. Advanced Algebra (Second Edition) [M]., Fudan University press,.2008[7], Wang Junqing. On the application of elementary transformation in Higher Algebra [J]. Journal of Cangzhou Teachers College,.2002.18 (3)[8], Li Haiyan, Wang Yanfang. The whole course learning guide of Linear Algebra (three edition of the National People's Congress). Dalian University of Technology press,.2008.8Application of [9] Guangyan. Linear algebra lecture. Dalian University of Technology press.2008.7[10] Northwestern Polytechnical University advanced algebra compilation group. Advanced algebra. Science Press,.2008[11] Zhao Lixin, once Wencai. Characteristics of square matrix with elementary transformation of matrix valued.2004 mathematics [J]. University[12] Zhan Hua Lu, Lu established. Some applications of elementary transformation of higher mathematics of.2006.11 [J].。

初等变换内容总结初等变换是线性代数中的重要概念，它是指通过一系列基本操作来改变矩阵的形态。

在本文中，我们将以人类的视角来描述初等变换的内容，并探讨其在实际问题中的应用。

一、初等变换的概念及基本操作初等变换是指通过三种基本操作对矩阵进行变换，这三种基本操作分别是：交换两行（列）的位置、某一行（列）乘以一个非零常数、某一行（列）的倍数加到另一行（列）。

这些操作可以改变矩阵的行列式、秩以及解的个数。

二、初等变换的应用初等变换在线性方程组的求解、矩阵的求逆以及线性相关性的判断等问题中都有广泛的应用。

下面我们将分别介绍这些应用。

1. 线性方程组的求解通过初等变换，我们可以将线性方程组转化为行阶梯形或简化行阶梯形矩阵，从而求解方程组的解。

通过交换行、乘以非零常数、行的倍数加到另一行等操作，我们可以将方程组转化为更加简单的形式，使得解的求解更加方便。

2. 矩阵的求逆通过初等变换，我们可以将一个方阵转化为单位矩阵，从而求得其逆矩阵。

逆矩阵在计算机图形学、电路分析等领域中有着重要的应用。

3. 线性相关性的判断通过初等变换，我们可以判断向量组的线性相关性。

通过将向量组转化为行阶梯形或简化行阶梯形矩阵，我们可以得到向量组的秩，从而判断其线性相关性。

三、初等变换的实例分析为了更好地理解初等变换的应用，我们将通过一个实际问题进行分析。

假设有一家电子公司生产A、B、C三种产品，每天生产的数量分别为a、b、c。

已知每个产品的销售价格分别为x、y、z，该公司每天的总收入为ax+by+cz。

现在，该公司决定调整产品的生产数量，以提高总收入。

通过初等变换，我们可以得到以下结论：- 如果将A产品的生产数量增加一个单位，总收入将增加x个单位。

- 如果将B产品的生产数量增加一个单位，总收入将增加y个单位。

- 如果将C产品的生产数量增加一个单位，总收入将增加z个单位。

基于以上分析，我们可以优化产品的生产方案，使得总收入最大化。

通过初等变换，我们可以得到一个线性规划问题，进一步求解出最优解。

多项式矩阵的初等变换多项式矩阵是指矩阵的元素是多项式的矩阵。

初等变换是矩阵运算中常用的一种方式，可以通过对矩阵进行一系列基本操作来改变其性质。

本文将介绍多项式矩阵的初等变换方法及其应用。

1.行交换：通过交换矩阵的两行来改变行的顺序。

例如，将第一行与第二行交换，可以写作R1<->R2。

2.行倍乘：将矩阵的某一行的所有元素乘以一个非零常数。

例如，将第一行的元素都乘以2，可以写作2R1。

3.行加减：将某一行的所有元素与另一行的对应元素相加或相减，然后替换该行。

例如，将第一行的元素分别加到第二行上，可以写作R2=R2+R1。

在实际问题中，多项式矩阵的初等变换被广泛应用。

以下是一些常见的应用场景：1.线性方程组的求解：通过初等变换，可以将线性方程组转化为简化的行阶梯形矩阵，从而求解出未知数的值。

2.矩阵的秩计算：通过初等变换，可以将矩阵转化为行阶梯形矩阵，进而计算出矩阵的秩。

3.矩阵的相似性判定：通过初等变换，可以将矩阵转化为标准型，从而判断两个矩阵是否相似。

4.多项式插值：通过初等变换，可以将多项式插值问题转化为线性方程组求解问题，从而得到多项式的系数。

多项式矩阵的初等变换是一种常用的矩阵运算方法，通过行交换、行倍乘和行加减操作，可以改变矩阵的性质以及解决实际问题。

在实际应用中，初等变换广泛用于线性方程组求解、矩阵的秩计算、矩阵相似性判定和多项式插值等领域。

熟练掌握多项式矩阵的初等变换方法对于数学问题的解决具有重要意义。

1.张广福，陈皓，袁启土.数学分析[M].高等教育出版社，2013.2.蒋继宗.线性代数与几何[J].高等教育出版社，2018.。

矩阵初等变换方法在高等代数中的应
用
矩阵初等变换在高等代数中有很多应用，下面列举部分内容：
- 求解线性方程组：通过初等变换可以将线性方程组转化为标准形式，更易于求解。

- 求矩阵和向量组的秩：通过初等变换不改变矩阵和向量组的秩，因此可以利用初等变换将矩阵或向量组化为标准形式，进而求出它们的秩。

- 化二次型为标准形：通过初等变换可以将二次型转化为标准形，更易于研究二次型的性质。

- 求一元多项式最大公因式：通过初等变换可以将一元多项式转化为标准形式，更易于求解一元多项式的最大公因式。

这些只是矩阵初等变换在高等代数中的部分应用，矩阵初等变换在高等代数中还有许多其他的应用，可以帮助我们解决各种问题。