矩阵的初等变换及其应用

石家庄经济学院本科生毕业论文

摘 要

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

ABSTRACT

Matrix 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;

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石家庄经济学院本科生毕业论文

目 录

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

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石家庄经济学院本科生毕业论文

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。这就使矩阵成为数学中一个应用广泛的概念，而作为矩阵的一种运算方法，初等变换在矩阵的研究中具有很重要的意义。

1 矩阵及其初等变换概念

首先介绍矩阵的概念 定义1.1 由m?n个数aij(i?1,2,?,m,j?1,2,?,n)排成m行n列的数表

?a11a12?a1n??a?a?a21222n? A??????????aa?am1m2mn??称为m行n列的矩阵，简称m?n矩阵。aij称为矩阵A的第i行第j列元素。

定义1.2 下面三种变换称为矩阵的初等行变换： (1)对掉矩阵两行（对调i,j两行，记作ri?rj）；

(2)用任意数k?0乘矩阵的某一行中的所有元素（第i行乘k，记作k?ri）； (3)用数k乘矩阵的某一行的所有元素加到另一行的对应元素上去（第j的k倍加到第i行上，记作ri?krj）。 把定义中的“行”换成“列”，就是矩阵的初等列变换的定义（所用记号是把r换成。 c）

矩阵的初等行变换和初等列变换统称为矩阵的初等变换。

定义1.3 对单位矩阵E施行一次初等变换得到的矩阵，称为初等矩阵或初等方阵。

?010??，将的第3行乘以-2100例如：我们将E的1，2两行互换得到初等矩阵E12??E????001???100??，将E的第1行的-5倍加到第2行上，就可以得到010倍所得初等矩阵E3(?2)??????00?2???100??。 E12(?5)???510????001??2 矩阵初等变换的应用

矩阵的初等变换应用广泛，本文主要总结了它在线性代数、数论、通信、经济、生物

基因遗传方面的应用。

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石家庄经济学院本科生毕业论文

2.1 在线性代数中的应用

矩阵的初等变换是矩阵的计算中必要的步骤。在矩阵计算时，首先需要对它进行初等变换，化成单位矩阵，阶梯形矩阵等简单的矩阵，使计算简便。 2.1.1 将矩阵化简为阶梯型和等价标准型

一个阶梯形矩阵，需满足两个条件：

（1）如果它既有零行，又有非零行，则零行在下，非零行在上。

（2）如果它有非零行，则每个非零行的第一个非零元素所在列号自上而下严格单调上升。

阶梯形矩阵的基本特征 ：

如果所给矩阵为阶梯形矩阵则矩阵中每一行的第一个不为零的元素的左边及其所在列以下全为零。

对于任何矩阵A，总可以通过有限次初等变换把矩阵化为阶梯形矩阵。 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：

?10?0?0??01?0?0???????????B??00?1?0?

?00?0?0???????????00?0?0??若B和A等价，称为矩阵A的标准形，主对角线上1的个数等于A的秩（1的个数可以是零）。

例 用初等变换将下列矩阵化为阶梯形和标准形，

?1131??1325??A???2267????2456?

解：应用初等变换

?1131??1131??1?1325??02?14??0??????A???2267??0005??0?????245602?14?????0?1000??1000??1?02?14??0200??0?????????0005??0005??0?????00000000?????0120001001??14??05??00?

00?00??10??00?3 2

石家庄经济学院本科生毕业论文

?1?0矩阵??0??0132?100001??1?04??是阶梯形，矩阵??05???0??0010000100?0??是标准形。 0??0?2.1.2 矩阵的分块和分块矩阵的初等变换

将分块乘法与初等变换结合就成为矩阵运算中重要的手段。 现设某个单位矩阵如下进行分块：

?Em0??0E?n? ?对它进行两行(列)对换；某一行(列)左乘(右乘)一个矩阵P；一行(列)加上另一行(列)的P (矩阵)倍数，就可得到如下类型的一些矩阵：

?0En??P0??Em0??EmP??Em0??E?,?0E?,?0P?,?0E?,?PE?,0n????n??n??m??

和初等矩阵与初等变换的关系一样，用这些矩阵左乘任一个分块矩阵

?AB??CD???, 只要分块乘法能够进行，其结果就是对它进行相应的变换：

?0Em??AB??CD????E?????n0??CD??AB?, (1) ?P0??AB??PAPB?, (2) ?0E??CD???C?D?n??????Em?P?0??AB??A?CD???C?PAEn?????B? 。 (3)

D?PB??同样，用它们右乘任一矩阵，进行分块乘法时也有相应的结果。

在(3)中，适当选择P，可使C?PA?0。例如A可逆时，选P??CA?1，则C?PA?0。于是(3)的右端成为

B?A??0D?CA?1B? ??这种形状的矩阵在求行列式、逆矩阵和解决其它问题时是比较方便的，因此(3)中的运算非常有用。 例 设

?A0??1TA,D。可逆，求。 T???CD???A0解：??CDE00??A0E???1E???0D?CA?E0????E??00EA?1?D?1CA?10? ?1?D?所以，T?1?A?1???1?1??DCA0?. ?1?D?3

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