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  • 拉普拉斯定理行列式乘法PPT学习教案.pptx
  • 行列式乘法规则的证明方法和应用毕业论文.doc
  • 本节定义行列式的一个新的性质,行列式的乘法性质 1 行列式的乘法性质 2 行列式乘法性质的证明 3 行列式乘法性质的几何意义

    本节定义行列式的一个新的性质,行列式的乘法性质

    0 行列式函数D的三条性质:

    (i)若存在ai=aj(i不等于j)则D=0

    (ii)D是多重线性函数

    (iii)D(e1,...,en) = 1

    1 行列式的乘法性质

    设A,B都是nxn的矩阵,则:
    det(BA) = det(A)det(B)

    2 行列式乘法性质的证明

    根据矩阵乘法的性质可知BA的第j列为(BA)ej(其中ej为单位矩阵E的第j列)

    同理A的第j列aj = Aej, 根据矩阵乘法的结合律有:

    (BA)ej=BAej=Baj

    于是根据上一节的内容可知 将 BA的行列式表示成函数形式有:


    假设detB 不等于 0 定义函数C:


    下边证明函数C满足函数D的三条性质(见之前的章节)则可以证明函数C是函数D本身

    (i) 如果存在ai = aj (i不等于j) 则Bai = Baj 因此D(Ba1,...,Ban) = 0 ,因此C(a1,...,an) = 0 因此函数C 满足性质1,

    (ii)Bai 是ai的线性函数, 又因为D是多重线性函数,因此C也是ai的多重线性函数(线性映射的复合仍是线性映射),因此函数C满足性质2

    (iii)若令ai = ei 则有:


    因此函数C满足性质3,综上所述函数C满足行列式函数的三条性质,因此函数C就是行列式本身 因此:


    3 行列式乘法性质的几何意义

    一个矩阵可以看成是一个线性映射那么线性映射将一个单形映射为另一个单形 体积变为原来的|detB| 倍 ,由于任意单形都是若干开集的并。因此任意开集在映射B下的象的体积

    (detBA)都是原体积(detA)的|detB|倍

    展开全文
  • 通过java实现矩阵的乘法行列式求值、伴随矩阵和逆矩阵的运算 在考研备考的过程中复习到线性代数时,发现行列式的求法多种多样,并且矩阵的变换也五花八门,有时候偶然的一个思路或许能发现知识点直接的联系,但是...

    通过java实现矩阵的乘法、行列式求值、伴随矩阵和逆矩阵的运算

    在考研备考的过程中复习到线性代数时,发现行列式的求法多种多样,并且矩阵的变换也五花八门,有时候偶然的一个思路或许能发现知识点直接的联系,但是如果每次都要为了验证自己的想法,并寻找规律时,花大量时间在行列式的计算上未免太过浪费,于是尝试写一些简陋的程序来减缓计算压力,同时练习自己的逻辑思维能力。
    最终代码如下:

    // 矩阵乘法
    	public static int[][] matrixmultiply(int[][] matrix1, int[][] matrix2) {
    		int m = matrix1.length;
    		int n1 = matrix1[0].length;
    		int n2 = matrix2.length;
    		if (n1 != n2) {
    			return null;
    		}
    		int s = matrix2[0].length;
    
    		int[][] result = new int[m][s];
    		int x = 0;
    		int y = 0;
    		for (x = 0; x < m; x++) {
    			for (y = 0; y < s; y++) {
    				int plus = 0;
    				for (int count1 = 0; count1 < n1; count1++) {
    					plus += matrix1[x][count1] * matrix2[count1][y];
    				}
    				result[x][y] = plus;
    			}
    		}
    
    		return result;
    	}
    
    	// 求逆矩阵
    	public static float[][] InverseMatrix(int[][] data) {
    		// 求矩阵的行列式值
    		int det = Determinantcount(data);
    		// 求伴随矩阵
    		int[][] adjoin = Adjointmatrix(data);
    		float[][] result = new float[data.length][data[0].length];
    		// 求逆
    		for (int i = 0; i < data.length; i++) {
    			for (int j = 0; j < data[0].length; j++) {
    				result[i][j] = ((float) adjoin[i][j] / (float) det);
    			}
    		}
    		return result;
    	}
    
    	// 伴随矩阵方法
    	public static int[][] Adjointmatrix(int[][] data) {
    		if (data.length != data[0].length) {
    			return null;
    		}
    		if (data.length == 1) {
    			return new int[][] { { 1 } };
    		}
    		int n = data.length;
    		int[][] result = new int[n][n];
    
    		// 分别求出data中aij的代数余子式值赋予result中的bji
    		for (int i = 0; i < n; i++) {
    			for (int j = 0; j < n; j++) {
    				// 求出aij的余子式并求值
    				int cofactorDet = Determinantcount(Cofactor(data, i, j));
    				if ((i + j) % 2 == 1) {
    					cofactorDet *= -1;
    				}
    				// 赋值bji
    				result[j][i] = cofactorDet;
    
    			}
    		}
    
    		return result;
    
    	}
    
    	// 求余子式的方法
    	public static int[][] Cofactor(int[][] data, int i, int j) {
    		int m = data.length;
    		int n = data[0].length;
    		if (m < i || n < j) {
    			System.out.println("越界");
    			return null;
    		}
    
    		int[][] result = new int[m - 1][n - 1];
    
    		for (int row = 0; row < m; row++) {
    			if (row == i) {
    				// 不做操作
    			} else if (row < i) {
    				for (int colu = 0; colu < n; colu++) {
    					if (colu == j) {
    						// 不做操作
    					} else if (colu < j) {
    						result[row][colu] = data[row][colu];
    					} else if (colu > j) {
    						result[row][colu - 1] = data[row][colu];
    					}
    				}
    			} else if (row > i) {
    				for (int colu = 0; colu < n; colu++) {
    					if (colu == j) {
    						// 不做操作
    					} else if (colu < j) {
    						result[row - 1][colu] = data[row][colu];
    					} else if (colu > j) {
    						result[row - 1][colu - 1] = data[row][colu];
    					}
    				}
    			}
    
    		}
    
    		return result;
    	}
    
    	// 矩阵行列式计算
    	public static int Determinantcount(int[][] data) {
    		int rowcount = data.length;
    		int columncount = data[0].length;
    		if (rowcount != columncount) {
    			return -1;
    		}
    		if (rowcount == 1) {
    			return data[0][0];
    		}
    		// 创建用于容纳余子式的容器
    		int[][] tempMatrix = new int[rowcount - 1][columncount - 1];
    		int plus = 0;
    
    		// 确认传入的行列式大于等于2阶,按第一行展开
    		for (int i = 0; i < columncount; i++) {
    
    			int a = data[0][i];
    			// 通过递归求第一行第i列的余子式的行列式值
    			for (int j = 0; j < columncount; j++) {
    
    				// 当i=j时
    				if (i == j) {
    				}
    				// 当i < j的情况
    				else if (j < i) {
    					for (int c = 0; c < rowcount - 1; c++) {
    						tempMatrix[c][j] = data[c + 1][j];
    					}
    				} else if (j > i) {
    					for (int c = 0; c < rowcount - 1; c++) {
    						tempMatrix[c][j - 1] = data[c + 1][j];
    					}
    				}
    
    			}
    			// 求代数余子式的-1项
    			if (i % 2 == 1) {
    				a *= -1;
    			}
    			int re = Determinantcount(tempMatrix);
    			// 累加展开项与代数余子式的乘积
    			plus += a * re;
    		}
    
    		return plus;
    
    	}
    

    由于练习题目中给出的大多是整数型矩阵,在矩阵乘法、求伴随矩阵和行列式值的方法中用的是整数型,只有求逆矩阵的运算用到了浮点型变量,涉及到的输入异常和越界异常也没怎么处理。

    展开全文
  • 行列式乘法定理

    千次阅读 2012-12-10 19:48:00
    方阵$A$和方阵$B$,可以在$A$和$B$之间做矩阵的乘法,得到方阵$|AB|$.则我们有 $$|AB|=|A||B|$$ 证明:为此我们先证明另一个命题,我们要证明必有 引理:$$|AB^T|=|A||B^T|$$ 下面用具体实例在说明该引理的正确...

    方阵$A$和方阵$B$,可以在$A$和$B$之间做矩阵的乘法,得到方阵$|AB|$.则我们有

    $$|AB|=|A||B|$$

     

    证明:为此我们先证明另一个命题,我们要证明必有

    引理:$$|AB^T|=|A||B^T|$$

    下面用具体实例在说明该引理的正确性.设

    \begin{equation}
    A=\begin{pmatrix}
    a_{11}&a_{12}&a_{13}\\
    a_{21}&a_{22}&a_{23}\\
    a_{31}&a_{32}&a_{33}\\
    \end{pmatrix}
    \end{equation}

    \begin{equation}
    B=\begin{pmatrix}
    b_{11}&b_{12}&b_{13}\\
    b_{21}&b_{22}&b_{23}\\
    b_{31}&b_{32}&b_{33}\\
    \end{pmatrix}
    \end{equation}
    下面我们来看$|A||B^T|$,$|A||B^T|$必定有$3!3!=36$项.下面我们来看其中一项.比方说其中一项是

    \begin{equation}
    a_{12}a_{21}a_{33}b_{13}b_{21}b_{32}
    \end{equation}

    该项的符号我们暂且不管它,稍后再考虑.


    我们现在来看,其实
    \begin{equation}
    a_{12}a_{21}a_{33}b_{13}b_{21}b_{32}
    \end{equation}在

    \begin{equation}
    |AB^T|
    \end{equation}里也是有出现的!而且是只出现一次的!在哪里有出现呢?我们来看

    \begin{equation}
    a_{11}b_{31}+a_{12}b_{32}+a_{13}b_{33}
    \end{equation}以及

    \begin{equation}
    a_{21}b_{21}+a_{22}b_{22}+a_{23}b_{23}
    \end{equation}以及

    \begin{equation}
    a_{31}b_{11}+a_{32}b_{12}+a_{33}b_{13}
    \end{equation}

    将这三者相乘,发现里面也有$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$.而且能拼凑出$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的也只有这三者.下面我们
    来表明,$|A||B^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$和$|AB^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的符号是相同的.这是一个神秘的事情.因此我将用一种神秘的语言来解释——一种形而上的语言.

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    根据这幅图,很容易辨别出$|A||B^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$和$|AB^T|$中的$a_{12}b_{21}a_{33}b_{13}b_{21}b_{32}$的符号是相同的.

     

     

    而且,容易得到$|AB^T|$中的每一项,$|A||B^T|$中也唯一地存在,且易得它们的符号是相同的.综上,可得

    $$|AB^T|=|A||B^T|$$

    引理得证.

     

    注:根据引理我们便很容易得知$|AB|=|A||B|$了.

    转载于:https://www.cnblogs.com/yeluqing/archive/2012/12/10/3827980.html

    展开全文
  • 线性代数中满足交换律的运算-行列式与迹

    线性代数的计算中往往乘法不满足交换律
    满足交换律的运算-行列式(determinant)与迹(trace)
    t r ( A T ) = t r ( A ) t r ( A + B ) = t r ( A ) + t r ( B ) t r ( A B ) = t r ( B A ) ( A B 为 n × n , B A 为 m × m ) tr(A^T)=tr(A)\\ tr(A+B)=tr(A)+tr(B)\\ tr(AB)=tr(BA)(AB为n×n,BA为m×m) tr(AT)=tr(A)tr(A+B)=tr(A)+tr(B)tr(AB)=tr(BA)(ABn×n,BAm×m)
    d e t ( A B ) = d e t ( A ) ∗ d e t ( B ) = d e t ( B A ) det(AB)=det(A)*det(B)=det(BA) det(AB)=det(A)det(B)=det(BA)
    行列式乘法可交换的初等变换证明:
    ( E B 0 E ) 为 列 初 等 便 换 矩 阵 ( A A B − E 0 ) = ( A O − E B ) ( E B 0 E ) ∣ A O − E B ∣ = ∣ A ∣ ∣ B ∣ ∣ A A B − E O ∣ = ( − 1 ) n 2 ∣ − E ∣ ∣ A B ∣ = ∣ A B ∣ \left( \begin{array} { l l } { E } & { B } \\ { 0} & { E} \end{array}\right)为列初等便换矩阵\\ \left( \begin{array} { l l } { A } & { AB } \\ { -E } & { 0} \end{array}\right)= \left( \begin{array} { l l } { A } & { O } \\ { -E } & {B} \end{array}\right) \left( \begin{array} { l l } { E } & { B } \\ { 0} & { E} \end{array}\right)\\ \left| \begin{array} { l l } { A } & { O } \\ { -E } & { B} \end{array}\right|=|A||B|\\ \left| \begin{array} { l l } { A } & {AB}\\{ -E } & { O } \\ \end{array}\right|=(-1)^{n^2}|-E||AB|=|AB|\\ (E0BE)便(AEAB0)=(AEOB)(E0BE)AEOB=ABAEABO=(1)n2EAB=AB

    矩阵的初等变换与"打洞"技巧

    展开全文
  • 行列式:每个单位正方形在变换后、变换前的面积——矩阵面积,例如: {1 0 0 1} 在欧式空间的直角坐标系下单位矩阵的面积是1。 {1 2 2 1} 的面积是: -3,如果按照几何意义来看: ...
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  • 行列式与矩阵的区别

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    1、行列式的本质是线性变换的放大率,而矩阵的本质就是个数表。 2、行列式行数=列数,矩阵不一定(行数列数都等于n的叫n阶方阵),二者的表示方式亦有区别。 3、行列式与矩阵的运算明显不同 (1) 相等:只有两个...
  • 行列式乘法及转置

    千次阅读 2019-05-06 15:11:48
    这里写#include
  • "请输入行列式的行数、列数\n" ); Dmt test; // int x; int y; scanf ( "%d%d" , &test.x, &test.y); printf ( "读取成功,请输入行列式\n" ); for ( int i = 1 ; i ; i++) { for ( int j = 1 ; j...
  • """ 递归(拉普拉斯展开)计算n阶行列式 传入的一定为"正方形" input: [ [1,2,3,4], [5,6,7,8], [9,1,2,3], [4,5,6,7], ...
  • 线性代数(一)-行列式

    千次阅读 2018-10-11 17:58:22
    一、二阶和三阶行列式 1.二阶行列式 PS:只适用于二元线性方程; 2.三阶行列式   二、全排列及其逆序数 1.全排列 把n个不同的元素排成一列,叫做这n个元素的全排列; 2.逆序数 对于n个不同的元素,先...
  • n阶行列式计算 方法:将大矩阵不断分解为小矩阵来缩小其规模,最后逐层返回答案。 java实现 import java.util.Scanner; class Main { public static void main(String []args) { Scanner sc=new Scanner...
  • 拉普拉斯定理和行列式乘法规则选讲PPT学习教案.pptx
  • 行列式乘法

    千次阅读 2013-10-08 19:40:06
    行列式det det(A*B)=det(A)*det(B);
  • 给定方阵 (AO) 的逆 (AI) 和行列式 (det) 可以通过以下方式直接找到[AI,det] = inv1(AO) 它使用自动旋转方案。 所有计算只涉及简单的矩阵乘法。 没有旋转的直接结果也可以通过AI = inv0(AO) 源代码只有 4 行语句。 ...
  • n阶行列式相乘

    2021-04-18 22:10:31
    n阶行列式相乘 关于行列式相乘 代码如下 #n阶行列式相乘(同阶)在实际中用的不是特变多,不如直接算n阶行列式 #无特殊情况处理 #N1表示第一个行列式 #M1表示第二个行列式 class Solution(object): def Ride(self):...
  • 理解矩阵乘法

    万次阅读 多人点赞 2019-05-10 20:22:00
    如果从线性方程的角度,理解矩阵乘法就毫无难度。 下面是一组线性方程。 矩阵的最初目的,只是为线性方程组提供一个简写形式。 老实说,从上面这种写法,已经能看出矩阵乘法的规则了:系数矩阵第一行的2和1,...
  • 矩阵乘法的几何理解

    2019-05-03 08:35:30
    矩阵乘法 对于一个向量aaaa=[−1,2]Ta=[-1,2]^{T}a=[−1,2]T当对向量aaa乘以一个矩阵BBB(4)B=[13−20] B=\begin{bmatrix} 1 &amp; 3 \\ -2 &amp; 0 \end{bmatrix} \tag{4} B=[1−2​30​](4) 即(4)[13−...
  • 作者证明了对于某个算术函数类,若 f 是一个半乘法函数且 属于这个函数类,则矩阵(f[xi,xj])是半正定的,进而给出了其行列式的明确的下界和上界。若以f(c)表示函数 f 的 c重狄利克雷乘积,则矩阵 也有类似的...
  • 行列式这个“怪物”定义初看很奇怪,一堆逆序数什么的让人不免觉得恐惧,但其实它是有实际得不能更实际的物理意义的,理解只需要三步。这酸爽~1,行列式是针对一个的矩阵而言的。表示一个维空间到维空间的线性变换。...
  • AB=C shape: A mxn , B nxp, C mxp 1、Cij=(row i of A)* (column j of B) 2、Matrix A times every column of B A[b1 ... bp]=[Ab1 ... Abp] 3、Every row of A times matrix B [row i of A] B= [row i......
  • Defining Matrix using Numpy 使用Numpy定义矩阵 Determinant of a Matrix 矩阵的行列式 Transpose Matrix 转置矩阵 Here, we will learn that the determinant of the transpose is equal to the m...
  • 行列式内部变换规则

    万次阅读 2020-03-27 17:41:27
    行列式的换法变换、倍法变换、消法变换为行列式的初等变换。 换法变换:交换两行(列)。 倍法变换:将行列式的某一行(列)的所有元素同乘以数k。 消法变换:把行列式的某一行(列)的所有元素乘以一个数k并加到...
  • 本问题中,将输入两个矩阵 A 和 B,实现对矩阵的数乘、矩阵加法、矩阵乘法以及行列式的计算。 如果对矩阵的算法不了解,请查阅相关资料。 【输入形式】 输入的第一行为两个正整数 M 和 N,分别表示矩阵 A 的...
  • 在前文 线性代数(1)—— 行列式中,我们已经对行列式有了比较直观的理解。 行列式最初用于表示线性方程组的系数,其值可以用于判别齐次线性方程组的解情况,也可以用于计算一般线性方程组的通解 可以从几何角度...
  • 矩阵/行列式的意义

    万次阅读 多人点赞 2017-02-24 16:34:24
    矩阵乘法规则看起来比较复杂,不容易理解其乘法规则背后隐含的意义。现举一个例子说明矩阵乘法的意义。如下图所示,一个商店出售Beef pie,chicken pie,vegetable pie,其单价分别为3元,4元,2元。此外,还统计出...
  • 行列式运算法则 矩阵的运算及其运算规则:

    万次阅读 多人点赞 2018-10-29 17:01:19
    1、三角形行列式的值,等于对角线元素的乘积。计算时,一般需要多次运算来把行列式转换为上三角型或下三角型 2、交换行列式中的两行(列),行列式变号(交换) 3、行列式中某行(列)的公因子,可以提出放到行列式...

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