2.3 矩阵的初等变换及其应用
矩阵初等变换及应用研究
矩阵初等变换及应用研究矩阵初等变换是线性代数中的一个基本概念，它是指对矩阵进行一系列的基本操作，包括交换两行（列），某行（列）乘k（k≠0），某行（列）乘k再加到另一行（列）上。
矩阵初等变换在线性代数中有广泛的应用，可以用来求解线性方程组、计算矩阵的秩和逆矩阵、求解特征值与特征向量等。
首先，矩阵初等变换可以用来求解线性方程组。
对于一个线性方程组，可以将其系数矩阵与增广矩阵进行同样的初等变换，从而化简方程组。
这样做的目的是为了找到一个等价的简化方程组，可以更方便地求解解集。
通过初等变换，可以将线性方程组化为行最简形式（也即梯形形），进而利用高斯-约当消元法或者矩阵的初等行变换求解线性方程组，得到唯一解、无解或无穷解。
其次，矩阵初等变换可以用来计算矩阵的秩和逆矩阵。
通过一系列的初等行（列）变换，可以将一个矩阵化为行最简形式（也即行阶梯形矩阵），从中可以直接读出矩阵的秩。
对于方阵，如果秩等于矩阵的阶数，则该矩阵可逆，可以利用初等变换求解逆矩阵。
逆矩阵的求解是矩阵初等变换的重要应用之一，通过应用矩阵初等变换，可以将一个方阵转化为单位矩阵，从而求出逆矩阵。
另外，矩阵初等变换还可以用来求解特征值与特征向量。
对于一个n阶方阵A，特征值一般通过求解方程det(A-λI)=0来求得，其中I是单位矩阵，λ是特征值。
通过初等行变换，可以将A-λI化为行最简形式，从而求解特征值。
特征值求解完毕后，可以利用矩阵初等变换求解对应的特征向量。
总结起来，矩阵初等变换是线性代数中的重要工具，广泛应用于求解线性方程组、计算矩阵的秩和逆矩阵、求解特征值与特征向量等方面。
通过一系列的基本操作，可以将矩阵化简为行最简形式，从而更方便地进行进一步的计算和分析。
矩阵初等变换的应用使得矩阵的求解和计算更加简便高效，提高了线性代数在实际问题中的应用能力。
矩阵初等变换及其在线性代数中的应用
矩阵初等变换及其在线性代数中的应用线性代数是一门重要的数学分支，它研究的是线性变换及其代数分析性质。
其中，矩阵是线性代数中非常重要的工具，它可以把线性方程组转化成一个更简单的形式，使得我们可以更容易地进行求解。
而矩阵的初等变换则是在求解线性方程组时必须要用到的一种基本技巧。
本篇文章将深入探讨矩阵初等变换及其在线性代数中的应用。
矩阵初等变换到底是什么？矩阵初等变换是指对于一个矩阵来说，可以通过三种基本变换操作得到新的矩阵。
这三种操作分别是：交换矩阵的任意两行或两列；用一个非零常数 k 乘以矩阵的某一行或某一列；将矩阵的某一行或某一列加上另一行或另一列的 k 倍。
这三种操作称为矩阵的行初等变换或列初等变换。
首先来看一个示例，假设有如下矩阵：$$\begin{bmatrix}1 &2 \\3 &4 \\\end{bmatrix}$$对于这个矩阵，我们可以进行如下初等变换：①交换第一行和第二行$$\begin{bmatrix}3 &4 \\1 &2 \\\end{bmatrix}$$②将第二行乘以2$$\begin{bmatrix}1 &2 \\6 & 8 \\\end{bmatrix}$$③将第二行减去第一行的两倍$$\begin{bmatrix}1 &2 \\4 & 4 \\\end{bmatrix}$$通过这三种基本变换，我们可以将原始矩阵变换成一个新的矩阵。
这个过程通常用矩阵的运算符号表示，比如将第二行减去第一行两倍的操作可以表示为：$$\begin{bmatrix}1 & 0 \\-2 & 1 \\\end{bmatrix}\begin{bmatrix}1 &2 \\3 &4 \\\end{bmatrix}=\begin{bmatrix}1 &2 \\1 & 0 \\\end{bmatrix}$$其中，左侧的矩阵就是一个变换矩阵，它表示了对原矩阵的操作。
矩阵的广义初等变换及应用
设 A, B, C , D ∈ M n ( F ) ,证明
A B C D B A D C C D A B D C B A
M =
=
1 8 2 0 −2 14 2 − 2 11 = 1 ⋅ 14 − 20 11 8 − ⋅ [0 − 20 2 2] −2 = 14 −5 = 118 − 24
−
1 B, 2
A 0
0 A B → B 0 B
→
A 0
A + B B
万方数据
芜湖职业技术学院学报 2005 年第 7 卷第 2 期
57
A 0 A A + B ∴ r ≥ r(A+B) 0 B =r 0 B
对此分块矩阵
则
A B C D
实施一次广义初等变换后得到的矩阵称为广义初等 矩阵 广义初等矩阵有下面三种形式 1
0 E n Em 0
B A 广义初等变换 → −1 0 D − CA B
由行列式的性质知在此变换过程中矩阵 M 作成的行 列式的值不变,即
→
−E E
− ( A − B) ( A − B) 1 1 [( A + B ) −1 − ( A − B ) −1 ] [( A + B ) −1 + ( A − B ) −1 ] 2 2
−1 −1
r(A)+r(B) ≤ n 证明 构造分块矩阵
E B E 0 E → → → B E A − AB 0 0 0 0 0
B −1 = 1 B B = 1 A − B −1 4 B − B 4
矩阵的初等变换及其应用
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。
2-3矩阵的初等变换
①
②
③ ④
1 2 1 1 2 1 1 1 2 3 1 1 6 9 7 3
r2 r3
r3 2r1 r4 3r1
4 2 B1 2 9
x1 x2 2 x3 x4 2x 2 2 x 3 2 x 4 5 x 2 5 x 3 3x 4 3 x 2 3 x 3 4 x4
定义：下列三种变换称为矩阵的初等行变换： 对调两行，记作 ri rj ； 以非零常数 k 乘某一行的所有元素，记作 ri k ； 某一行加上另一行的 k 倍，记作 ri krj . 其逆变换是：
ri rj
ri k
ri rj ;
ri k;
初等变换
初等行变换 初等列变换
备注
带有运算符的矩阵运算，用“ = ”．例如： 矩阵加法 ＋ 数乘矩阵、矩阵乘法 × 矩阵的转置 T（上标） 方阵的行列式 |∙| 不带运算符的矩阵运算，用“～”．例如： 初等行变换 初等列变换
二、矩阵之间的等价关系
r
行等价，记作 A ~ B
A
有限次初等行变换 有限次初等列变换
B
c
列等价，记作 A ~ B
① ② ③ ④
1 0 0 0
1 2 1 4 1 1 1 0 B4 0 0 1 3 0 0 0 0
x1 x2 2 x3 x4 4, x2 x3 x4 0, x4 3, 0 0.
①
②
③ ④
③ ④
④－2×③
4, ① 0, ② 6, ③ 3. ④
1 0 0 0
r3 r4
矩阵的初等变换及其应用
线性代数第一次讨论课1;要求2;正文3;个人总结丁俊成00101209第一部分：要求线性代数课程的主要任务是夯实工程问题的数学基础，培养学生的逻辑思维、定量分析、数学建模、科学计算的数学能力，提高数学素养。
讨论课是以学生为主导，其内容包括理论内容的专题讨论、探究性应用案例的数学模型的建立。
通过对理论内容的深入探讨，加深学生对知识的深刻理解与掌握，培养学生自主学习能力、逻辑思维能力、对知识的归纳梳理与综合能力，提高学生分析问题与数学建模的能力。
第一次讨论课内容矩阵初等变换及其应用请卓越班的同学们按照下面的提纲（内容包括概念、求解方法、举例、应用案例等）准备。
要求做成word或PPT文档。
同学们自荐或推荐上讲台讲课。
希望同学们踊跃参与。
第一次讨论课的时间初步定在5月中旬。
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）第二部分：正文矩阵的初等变换及其应用矩阵是线性代数最基本也是最重要的概念之一，几乎线性代数所有的概念或者其使用里面都可以见到矩阵的身影，作为矩阵核心，矩阵的初等变换及其应用是及其重要的，本文将对矩阵初等变换及其应用做简单讨论。
一．两个矩阵的等价矩阵等价的定义为：若矩阵A经过一系列初等行变换化为矩阵B，则称A与B行等价。
若矩阵A经过一系列初等列变换化为矩阵B，则称A与B列等价。
若矩阵A经过一系列初等变换化为矩阵B，则称A与B等价（相抵）。
根据性质，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。
矩阵的初等变换
矩阵的初等变换及其应用作者（学号）：李婷婷（20072238）学校名称：皖西学院系别：应用数学学院专业：信息与计算科学班级：0701班指导老师：岳芹二○一一年六月矩阵的初等变换及其应用作者李婷婷指导老师岳芹摘要: 本文从矩阵的初等变换的概念出发,以具体实例为依据,总结了矩阵初等变换在线性代数中的一些应用.可以用来求逆矩阵、求矩阵的秩、求向量组的极大无关组、证明向量组等价,判断向量组的线性相关性、解矩阵方程和化二次型为标准形等.另外,简单介绍了矩阵的初等变换在其他方面的应用.关键词:矩阵；初等变换；应用The elementary transformation of matrices and its applications Abstract: Starting with the concept of the elementary transformation of matrices summarizes , based on examples, applications of the elementary transformation in liner algebra ar e summarized. It can be used to find inverse matrices, rank of a matrix and enormous liner independence group of a class of vectors, and prove the equation of vector groups, judge the linear independence of a vector group, solve matrix equations and change a quadratic form from quadratic form to standard form and so on. In addition, applications of the elementary transformation of matrices in other aspects is simply introduced.Key words: matrix; elementary transformation; application1 引言在数学名词中,矩阵（英文名Matrix）是用来表示统计数据等方面的各种有关联的数据,这个定义很好地解释了Matrix代码是制造世界的数学逻辑基础.数学上,矩阵是指纵横排列的二维数据表格,最早来自于方程组的系数及常数所构成的方阵,这一概念由19世纪英国数学家凯利首先提出.矩阵是数学中的一个重要内容,是线性代数核心,在自然科学、工程技术和经济领域都有广泛的应用.矩阵的初等变换是矩阵中一种十分重要的运算,它在解线性方程组、求逆矩阵及矩阵理论的探讨中都可起到非常重要的作用.由于矩阵的初等变换计算简洁,便于应用,是研究代数问题的一个重要工具.如何巧妙地运用初等变换去解决数学中有些运算复杂的问题会起到事半功倍的效果.本文将对矩阵的初等变换在线性代数中的若干应用进行了简要讨论,首先给出了矩阵初等变换的定义,然后对其相关的各方面的应用，结合具体实例进行总结。
2.3 矩阵的初等变换与初等矩阵
~
3 0 2 0 1 0 0 2 1 1 0 0 0 9 4 0 2 3
3 0 2 0 1 0 ~ 0 2 1 1 0 0 r3 9 r2 0 0 1 9 4 6 3 0 0 18 9 12 r1 2 r3 0 2 0 8 4 6 ~ r2 r3 0 0 1 9 4 6
4 1 2 1
00 00 11 00
0 0 10 20 30 00 00 00 00
9 4 6 0 0 0 2 0 8 3 0 00
矩 阵 A 的 标 准 型
例4.2
设
1 1 2 1 A 1 1 1 0 2 0 1 1
的等价标准形.
求
A
注:
1.任一矩阵都可经过初等行变换化成行阶梯矩阵； 2.任一矩阵都可经过初等行变换化成行最简矩阵；
3.任一矩阵都可经初等变换r
Er 0, E r 都是 0
0 的特殊情况. 0
O Er 。 O O
行阶梯形矩阵
也就是指可以画一条阶梯折线，
折线的下方元素全为零；并且每个阶梯只有一行，
阶梯数即为非零行的行数，阶梯线每一竖线后面第
一个元素为非零元.
3 3 2 1 0 1 0 , B 0 0 1 2 5 如: A 0 0 0 0 0 6 0 1 1 0 0 0 8 0 0 2 5 0 0 5 2 4 0 2 1 0 4 , C 0 3 0 0
0 1 1 3 0 0 0 0 0 2 0 0 0 2 0 8 1 3 0 0
为行阶梯矩阵.
行最简形矩阵
是指行阶梯形矩阵中除每一竖线后面的第一个
矩阵的初等变换及应用(吴礼斌)
对 B 进一步化为行简化矩阵
3. 求逆矩阵
版权所有，安徽财经大学统计与应用数学学院吴礼斌，13955236046
2
线性代数
0 1 1 设矩阵 A = 1 1 2 ，求 A −1 。 2 −1 0
解：A 是 3 阶矩阵，在 A 的右边写上 3 阶单位矩阵，并对其施行初等行变换，得
版权所有，安徽财经大学统计与应用数学学院吴礼斌，13955236046 5
线性代数
其中 c1 , c 2 为任意常数。 （2）求解齐次线性方程组
x1 + x2 + x3 + x4 + x5 = 0, 3x + 2 x + x + x − 3x = 0, 1 2 3 4 5 5 x1 + 4 x2 + 3x3 + 3x4 − x5 = 0, x2 + 2 x3 + 2 x4 + x5 = 0.
再由行简化形矩阵写出原方程组的同解方程组为
x1 − 2 x2 − 2 x4 = −4 +1 x =5 2 4 2 x3
移项得
x1 = −4 + 2 x 2 + 2 x 4 5 −1 x3 = 2 2 x4
令 x2 = c1 , x4 = c2 ，代入上面同解方程组得原方程组的通解（一般表示形式）为
线性代数
矩阵的初等行变换及应用
一、矩阵的初等行变换概念
定义。 初等行 定义。对矩阵进行下列三种变换，称为矩阵的初等 初等行变换。 变换 (1)交换矩阵某两行的位置； (2)用一个非零数乘以矩阵某一行的每一个元； (3)将矩阵某一行的元都乘以数 λ 后对应加到另一行上. 并称(1)为换法行变换，称(2)为倍法行变换，称(3)为倍加行变换. 若把对矩阵施行的三种“行”变换改为对“列”的三种变换，称为矩阵的初等列 变换。矩阵的初等行变换和初等列变换统称为矩阵的初等变换 初等变换。 初等变换。 为了表示的方便，我们引入如下的一组变换运算符号： ri ↔ rk 表示交换矩阵的第 i 行与第 k 行的位置；
矩阵的初等变换及应用的总结
矩阵的初等变换及应用内容摘要：矩阵是线性代数的重要研究对象。
矩阵初等变换是线性代数中一种重要的计算工具，利用矩阵初等变换，可以求行列式的值，求解线性方程组，求矩阵的秩，确定向量组向量间的线性关系。
一矩阵的概念定义：由于m×n个数aij（i=1，2，….，m；j=1，2，….，n）排成的m行n列的数表，称为m行n列，简称m×n矩阵二矩阵初等变换的概念定义：矩阵的初等行变换与初等列变换，统称为初等变换1.初等行变换矩阵的下列三种变换称为矩阵的初等行变换:(1) 交换矩阵的两行(交换两行,记作);(2) 以一个非零的数乘矩阵的某一行(第行乘数,记作);(3) 把矩阵的某一行的倍加到另一行(第行乘加到行,记为).1.初等列变换把上述中“行”变为“列”即得矩阵的初等列变换3 ,如果矩阵A经过有限次初等变换变成矩阵B,就称矩阵A 与矩阵B等价，记作A~B矩阵之间的等价关系具有下列基本性质:(1) 反身性;(2) 对称性若,则;(3) 传递性若,,则.三矩阵初等变换的应用1.利用初等变换化矩阵为标准形定理：任意一个m× n矩阵A，总可以经过初等变换把它化为标准形2.利用初等变换求逆矩阵求n阶方阵的逆矩阵：即对n×2n矩阵（A¦E）施行初等行变换，当把左边的方阵A变成单位矩阵E的同时，右边的单位矩阵也就变成了方阵A的逆矩阵A^(-1)即(A|E)经过初等变换得到(E|A^(-1))这种计算格式也可以用来判断A是否可逆，当我们将A化为行阶梯形矩阵时，若其中的非零行的个数等于n时，则A可逆，否则A不可逆。
设矩阵可逆，则求解矩阵方程等价于求矩阵，为此，可采用类似初等行变换求矩阵的逆的方法，构造矩阵，对其施以初等行变换将矩阵化为单位矩阵，则上述初等行变换同时也将其中的单位矩阵化为，即.这样就给出了用初等行变换求解矩阵方程的方法.同理, 求解矩阵方程等价于计算矩阵亦可利用初等列变换求矩阵. 即.3．利用矩阵初等变换求矩阵的秩矩阵的秩的概念是讨论向量组的线性相关性、深入研究线性方程组等问题的重要工具.从上节已看到，矩阵可经初等行变换化为行阶梯形矩阵，且行阶梯形矩阵所含非零行的行数是唯一确定的, 这个数实质上就是矩阵的“秩”,鉴于这个数的唯一性尚未证明，在本节中，我们首先利用行列式来定义矩阵的秩，然后给出利用初等变换求矩阵的秩的方法.定理：矩阵的初等变换不改变矩阵的秩，即若A~B则R（A）=R(B)为求矩阵的秩，只要把矩阵用初等行变换变成阶梯矩阵解体矩阵中非零行的行数即是该矩阵的秩利用矩阵值得概念，能够讨论线性方程组有解的条件，然后通过研究向量组的线性相关性，向量组的秩等重要概念，讨论线性方程组的结构。
矩阵的初等变换及其应用(Elementary transformation of matrix and its application)
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].。
2.3 初等矩阵与逆矩阵
同理可得
n A A A E . * A A Aki akj ij ij k 1
定理1 矩阵 A 可逆的充要条件是 A 0 ，且 1 1 A A, A
其中A为矩阵A的伴随矩阵.
1 1 A 即有 A 使 AA E. 证明 若 可逆，
当 A 0时, A称为奇异矩阵,当 A 0时, A称为 非奇异矩阵.
由此可得A是可逆阵的充要条件是 A为非奇异矩阵.
一、填空题（本题15分，每小题3分） ( 2010级考题)
1 9 2、设A为三阶方阵，且 A = 3，则 | ( A ) 1 | _____ .
A A 提示 AA A A A E A A E, A A
1、 对调两行或两列 对调 E 中第 i , j 两行，即 ( ri rj )，得初等方阵
1 1 0 1 第i 行 1 E (i , j ) 1 1 0 第 j 行 1 1
所以
0 1 A . 1 2
1
定义 行列式 A 的各个元素的代数余子式Aij 所 构成的如下矩阵
A11 A12 A A 1n
性质
A21 An1 A22 An 2 A2 n Ann
称为矩阵 A 的伴随矩阵.
按逆矩阵的定义得
A A 1 1 1 （A ） （A ） = 3 A= 2. A A A A
1
推论 证明
若AB E 或BA E , 则B A1 .
A B E 1,
故 A 0,
因而A1存在, 于是
矩阵的初等变换及其应用（Elem...
矩阵的初等变换及其应用（Elem...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, application Catalog1. 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 linearalgebra 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 2.3matrices1, 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 equivalent 3, 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 simpleformula 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<="" there="">(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, theunknown 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 matrix Solution: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 "two times".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 usingthe 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 elementarytransformation 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,.2008.8 Application 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].。
第二章 矩阵代数 S3_2矩阵的初等变换
1
i列
j列
1
i列
j列
i行
0
1
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j行
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因此： 类似可得:
(Eij )1 Eij
初等矩阵是可
( Eii
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Eii
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0)
逆矩阵，而且 它们的逆矩阵
(Eij (k))1 Eij (k)
也是初等矩阵.
16
几个定理性结论
1. 矩阵A与B等价
有初等矩阵
0
0
mn
17
4. n 阶矩阵A为可逆的 等矩阵的乘积
A Q1Q2 Qt .
它能表成一些初
5. 可逆矩阵总可以经过一系列初等行变换化成单 位矩阵.
【可逆矩阵总可以经过一系列初等列变换化成 单位矩阵.】
18
三、用初等变换求逆矩阵
设An可逆，则存在一系列初等矩阵 P1 ,
使
E Pm P1 A
所以 于是 Pm
1 2
1 0
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1
2
2
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5 1
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5 1 1 5 2 1
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1
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A A|
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2 0 1
2
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30
§2.4.1转置矩阵
矩阵的初等变换与应用
矩阵的初等变换与应用09金融2班 王启会 2009241078一、矩阵概念线性方程组 的解取决于 系数 常数项线性方程组的系数与常数项按原位置可排为这就是矩阵。
矩阵的定义 由m ×n 个数 排成的m 行n 列的数表称为m 行n 列的矩阵，简称m ×n 矩阵。
记作这m ×n 个数称为矩阵A 的元素，简称为元，数称为矩阵A 的（i,j ）元。
以数 为（i,j ）元的矩阵可记作 或 ，m ×n 矩阵A 也记作 元素是实数的矩阵称为实矩阵，元素是复数的矩称为复矩阵。
行数与列数都等于n 的矩阵称为n 阶矩阵或n 阶方阵.n 阶矩阵 A 也记作 只有一行的矩阵 称为行矩阵，又称行向量。
只有一列的矩阵 称为列矩阵，又称列向量。
注意：1.矩阵是数表，行列式是由其元素经适当定义一种运算而得到的数。
2.矩阵中行数与列数可以相等，也可以不相等。
而行列式中的行数与列数必须相等。
两个矩阵的行数相等，列数也相等时，就称它们为同型矩阵。
如果 与 是同型矩阵，并且它们的对应元素相等，即 那么就称矩阵A 与矩阵B 相等。
记作A =B 。
元素都是零的元素称为零矩阵，记作0。
),,2,1;,,2,1(n j m i a ij ==mn m m n n a a a a a a a a a212222111211⎪⎪⎪⎪⎪⎭⎫⎝⎛=mn m m n n a a a a a a a a a A 212222111211⎪⎪⎩⎪⎪⎨⎧=+++=+++=+++n n nn n n n n n n b x a x a x a bx a x a x a b x a x a x a 22112222212111212111⎪⎪⎪⎪⎪⎭⎫⎝⎛n nn n n n n b a a a b a a a b a a a 21222221111211ij a ij a n m ij a ⨯)()(ij a .n m A ⨯.n A )(21n a a a A =⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=n b b b B 21),,2,1;,,2,1(n j m i b a ij ij ===系数 (),,,2,1,n j i a ij =()n i b i ,,2,1 =)(ij a A =)(ij b B =二、矩阵的初等变换的定义1.定义矩阵的初等变换：下面的三种变换称为矩阵的初等变换 (1) j i r r ↔；j i c c ↔.（换行或换列） (2) k r i ⨯；k c i ⨯（数0≠k ）（倍行或倍列） (3) j i kr r +；j i kc c +..（倍行加或倍列加）2.矩阵A 与B 等价：n m A ⨯经过有限次的初等变换变成B . 记作~A B . （1）等价的性质：反身性 ~A A ；对称性 若~A B ，则~B A ；传递性 若~,~A B B C ，则~A C .（2）任何矩阵n m A ⨯都等价于一个标准形矩阵,即()()()()()r r r n r m n m n n r r n r n r E O A F r O O ⨯⨯-⨯⨯-⨯-⨯-⎛⎫→=⎪⎝⎭．即存在有限个初等矩阵l P P P ,,,21 , 使121()k k l m n PP P AP P F r +⨯= .且矩阵的等价标准形惟一确定.（3）行阶梯矩阵：可画出一条阶梯线，线的下方全是零；每个台阶只有一行，台阶数为非零行的行数，阶梯线的竖线（每段竖线的长度为一行）后面的第一个元素为非零元，即非零行的第一个非零元.例如1210104112140110301113,00013000130000000000A A --⎛⎫⎛⎫ ⎪ ⎪-- ⎪ ⎪==⎪⎪-- ⎪ ⎪⎝⎭⎝⎭上述两矩阵均为行阶梯矩阵.（4）行最简形矩阵：非零行的非零首元为1，且这些非零元所在的列的其他元素都为零的行阶梯矩阵.110104011030001300000A -⎛⎫ ⎪- ⎪= ⎪- ⎪⎝⎭为行最简形矩阵. 例1 求所给矩阵A 的行阶梯矩阵、行最简形矩阵以及等价标准型矩阵.123221112112141121421112 ~46224231123697936979r r r A ↔÷---⎛⎫⎛⎫ ⎪ ⎪---- ⎪ ⎪= ⎪ ⎪---- ⎪ ⎪--⎝⎭⎝⎭4221323141r 222311214112140331603316 ~ 05536011160334300039~r r r r r r r r r +------⎛⎫⎛⎫⎪ ⎪------ ⎪ ⎪ ⎪ ⎪----- ⎪ ⎪---⎝⎭⎝⎭2333243r 3(4)311214112140111601116000412000130003900000~~r r r r r r +÷-↔---⎛⎫⎛⎫⎪⎪---- ⎪ ⎪⎪ ⎪-- ⎪⎪-⎝⎭⎝⎭（行阶梯矩阵）1232321010401103~0001300000r r r r r --+-⎛⎫ ⎪- ⎪ ⎪- ⎪⎝⎭315132525434433451000001000 ~0010000000c c c c c c c c c c c c F +-+++⨯↔⎛⎫⎪⎪= ⎪⎪⎝⎭.（行最简形矩阵） （等价标准型矩阵）3.初等矩阵的概念（1）定义初等矩阵：由单位矩阵E 只经过一次初等变换得到的方阵. ①j i r r ↔或j i c c ↔ 均对应初等方阵),(j i E :行第行第j i j i E ←←⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=1101111011),(②k r i ⨯或k c i ⨯ )0(≠k 均对应初等矩阵))((k i E :行第i k k i E ←⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=1111))((③j i kr r +或i j kc c + 均对应初等矩阵))(,(k j i E :行第行第j i k k j i E ←←⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭⎫ ⎝⎛=1111))(,(（2）初等矩阵行列式的性质(,)1;(())(0);(,())1E i j E i k k k E i j k =-=≠= .重要结论：初等矩阵是可逆矩阵，且逆矩阵仍然是初等矩阵.(3)初等矩阵的逆矩阵①),(),(1j i E j i E =-;② ))1(())((1ki E k i E =-, )0(≠k ;③))(,())(,(1k j i E k j i E -=-. (4)初等矩阵的转置也是初等矩阵.①),(),(j i E j i E T =;② ))(())((k i E k i E T =, )0(≠k ; ③))(,())(,(k i j E k j i E T =.4.矩阵初等变换的重要性质【性质1】 设A 是一个m n ⨯的矩阵，对A 实施一次初等行(列)变换， 相当于在A 的左边(右边)乘以相应的m 阶（n 阶）初等矩阵. 【性质2】 方阵m m A ⨯可逆的充要条件是存在有限个初等矩阵12,,,n P P P ，使得12n A PP P = ，即~rA E .【定理】设A 与B 为m n ⨯矩阵，则①~rA B ⇔存在m 阶可逆矩阵P ,使PA B =. ②~c A B ⇔存在n 阶可逆矩阵Q ,使AQ B =.③~r c A B ⇔分别存在m 、n 阶可逆矩阵P 、Q ,使PAQ B =.5.用初等变换求逆矩阵或解矩阵方程的方法①若A 可逆,则1-A 也可逆,于是存在初等矩阵12,,,n P P P ,使12n A PP P = ，又1AAE -=即112n PP P A E -= ，所以111121n A P P P ----= , 用分块矩阵运算表示为111(,)(,)A A E A A A ---=⇔ 1(,)~(,)rA E E A -.1111~C A A E AA A E E A A ----⎛⎫⎛⎫⎛⎫⎛⎫=⇔ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭. ②用初等变换求解矩阵方程，求解线性方程组(1)解矩阵方程AX B =，其中A 可逆，则1X A B -=即 111(,)(,)(,)~(,)rA AB E A B A B E A B ---=⇔. (2)解线性方程组Ax b =，其中A 可逆.则1x A b -=，即 1(,)~(,)rA b E A b -.(3)解矩阵方程XA B =，其中A 可逆，则1X BA -=即 111~c A E A E A B BA B BA ---⎛⎫⎛⎫⎛⎫⎛⎫=⇔ ⎪ ⎪⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭. 【定理6】 矩阵方程 AX B =有解的充要条件是 ()(,)R A R A B =.例2设1221311122,2,013225A b b --⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪=-== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭，求线性方程组 12,Ax b Ax b ==的解. 解 设1212(,),(,)X x x B b b ==12211231321110122220(,,)1222033113512205005~r r r r A b b r r ↔⎛⎫⎛⎫---- ⎪ ⎪=---- ⎪ ⎪+ ⎪ ⎪--⎝⎭⎝⎭322123320001222142522000000111130030031212~~r r r r r r r r ↔⎛⎫⎛⎫--÷-+ ⎪ ⎪ ⎪ ⎪+ ⎪ ⎪--⎝⎭⎝⎭()1112,,E A b A b --=. 因为~r A E ，所以A 可逆，且1420132X A B -⎛⎫- ⎪== ⎪ ⎪-⎝⎭，即线性方程组12,Ax b Ax b ==都有惟一解，且解依次为111122420,132x A b x A b ---⎛⎫⎛⎫ ⎪ ⎪==== ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭.3.矩阵的秩（1）定义矩阵A 的k 阶子式：在n m ⨯矩阵A 中，任取k 行与k 列， 位于这些行列相交处的2k 个元素，按原相对位置构成的k 阶行列 式.（{}1min ,k m m ≤≤）kk k k kkk kj i j i j i j i j i j i j i j i j i i i i j j j a a a a a a a a a A 2122212121112121)(=.A 的k 阶子式共有kn k m C C 个. 例3 矩阵⎪⎪⎪⎭⎫ ⎝⎛-=121021121021A 的k 阶子式： (1) 1阶子式如：12()A ：2,共有121413=C C 个.(2) 2阶子式如：1324()A ：2111-,共有182423=C C 个. (3) 3阶子式如：123124()A ：121212011-,共有43433=C C 个. （2）定义矩阵A 的秩——设矩阵A 中有一个非零的r 阶子式D ,而且所有1+r 阶子式(如果存在的话)值全为0，则D 称为矩阵A 的最高阶非零子式，数r 称为矩阵A 的秩,记作()R A ,即()R A r =.注：①零矩阵的秩规定为0.②A 的最高阶非零子式称为矩阵A 的秩子式.例4 显然矩阵⎪⎪⎪⎭⎫ ⎝⎛=000000100001A 的秩为2；(())m n R F r r ⨯=.(3)矩阵秩的性质①0()min(,)m n R A m n ⨯≤≤.（结论显然成立）②若n A 可逆,则()n R A n =（也称非奇异矩阵或满秩矩阵）.此时0A ≠. 若n A 不可逆,()n R A n <，即方阵是降秩矩阵（也称为奇异矩阵）.此时有0A =(注意：降秩与满秩矩阵都是对方阵而言的) ③()()T R A R A =.④初等变换不改变矩阵的秩,即()()()R PA R AQ R A ==，其中Q P , 为初等矩阵. 若Q P ,均可逆,则()()()()R PA R AQ R PAQ R A ===. 若~A B ,则()()R A R B =.⑤max{(),()}(,)()()R A R B R A B R A R B ≤≤+. ⑥()()()R A B R A R B +≤+.⑦若0m n n l A B ⨯⨯=,则()()R A R B n +≤.结论：①将一个矩阵左乘一个列满秩矩阵时，其秩不变.②将一个矩阵右乘一个行满秩矩阵时，其秩不变. ③矩阵的初等行变换不改变秩子式的列位置；矩阵的初等列变换不改变秩子式的行位置.二、例题1、解方程 010100143100001201001010120X -⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪=- ⎪ ⎪ ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝⎭.解 因为010100100(1,2),001(2,3)001010E E ⎛⎫⎛⎫ ⎪ ⎪== ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭，且11(1,2)(1,2),(2,3)(2,3)E E E E --==，故方程的解为 11143(1,2)201(2,3)120X E E ---⎛⎫⎪=- ⎪ ⎪-⎝⎭143(1,2)201(2,3)120E E -⎛⎫ ⎪=- ⎪ ⎪-⎝⎭ 201210143(2,3)134120102E --⎛⎫⎛⎫⎪ ⎪=-=- ⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭.2、设A 为3阶矩阵，将A 的第2行加到第1行得B ，再将B 的第1列的1-倍加到第2列得C ，记110010001P ⎛⎫ ⎪= ⎪ ⎪⎝⎭，则（ ）（Ａ）1C P AP -=. （Ｂ）1C PAP -=. （Ｃ）TC P AP = （Ｄ）TC PAP =. 【分析】利用矩阵的初等变换与初等矩阵的关系以及初等矩阵的性质可得.【详解】由题设可得11011011011010,010010010001001001001B A C B A --⎛⎫⎛⎫⎛⎫⎛⎫⎪ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭， 而 1110010001P --⎛⎫⎪= ⎪ ⎪⎝⎭，则有1C PAP -=.故应选（Ｂ）.3、设n 维向量0,),0,,0,(<=a a a T α；E 为n 阶单位矩阵，矩阵TE A αα-=，T aE B αα1+=，其中A 的逆矩阵为B ，则a=______.【分析】 这里Tαα为n 阶矩阵，而22a T =αα为数，直接通过E AB =进行计算并注意利用乘法的结合律即可. 【详解】 由题设，有)1)((T Ta E E AB αααα+-= =TT T T a a E αααααααα⋅-+-11=TT T T a a E αααααααα)(11-+-=TT T a a E αααααα21-+-=E aa E T=+--+αα)121(,于是有 0121=+--a a ，即 0122=-+a a ，解得 .1,21-==a a 由于01a a <⇒=-.4、设三阶矩阵⎪⎪⎪⎭⎫ ⎝⎛=a b b b a b b b a A ，若A 的伴随矩阵的秩为1，则必有( ).(A) a b =或02=+b a . (B) a b =或20a b +≠.(C) a ≠b 且02=+b a . (D) a ≠b 且20a b +≠. 【分析】 A 的伴随矩阵的秩为1, 说明A 的秩为2，由此可确定a,b 应满足的条件. 【详解】 根据A 与其伴随矩阵A *秩之间的关系知，()2R A =，故有0))(2(2=-+=⎪⎪⎪⎭⎫ ⎝⎛b a b a a b b b a b b b a ，即有02=+b a 或a b =. 当a b =时，显然()2R A ≠, 故必有a b ≠且02=+b a . 应选(C).5、设n 阶矩阵A 与B 等价, 则必有( )(A) 当)0(||≠=a a A 时, a B =||. (B) 当)0(||≠=a a A 时, a B -=||. (C) 当0||≠A 时, 0||=B . (D) 当0||=A 时, 0||=B . 【分析】 利用矩阵A 与B 等价的充要条件: )()(B r A r =立即可得.【详解】因为当0||=A 时, n A r <)(, 又 A 与B 等价, 故n B r <)(, 即0||=B , 故选(D).。