第一讲矩阵基本运算

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,MATLAB,入门,1. 双击图标，进入,Matlab,界面（,command),2. 单击,file New Mfile,进入编辑界面(,Untitled1) ,进行编程之后，点击保存时可以修改文件名,（,姓名英文缩写）,3.要显示运算的结果，有两种方法：,(1)进入,command,界面，健入你定义的文件名，然后按回车键即可得到计算结果；,(2)点击编辑界面上方,Debug Run,于是运行结果出现在,command,界面。,Mat lab,一. 矩阵与向量的基本运算,1.矩阵（向量、数组）的输入方法,矩阵的输入利用 ，采取分行输入方法，,每个元素之间用逗号或空格，每行之间用分号.,例1.矩阵,A=,Matlab,输入：,A=1,5,1,0,1;2,6,0,1,1;3,7,1,0,1;4,8,0,1,1;,注：; 分号的作用在于运算结果不显示.,n,维行(列)向量可以看成是一个行(列) 矩阵，因此向量的输入和矩阵一样.,2.矩阵的合成与分解,例2.矩阵,A=,求,A,的第一行与第一列,解：,A,1,=,A,(1,:),表示矩阵,A,的第一行；,A,2,=,A,(:,1),表示矩阵,A,的第一列；,练习：,A,(4,:)，,A,(3,2)，,分别表示什么？,例3. 求矩阵,A,的第1,3,4行元素组成的矩阵.,解：首先健入,a,=1,3,4;,然后健入,B=A,(,a,:),即可,其中,a,=1,3,4,称为,索引向量,.,练习：求矩阵,A,的第1,3,4列元素组成的矩阵,例4. 求从矩阵,A,中去掉第1,2列后，剩余元素组成的矩阵.,解：,B=A,(:,3:5);,注意：3:5 表示从3开始按步长为1 增加到5.,B = 1 0 1,0 1 1,1 0 1,0 1 1,练习：从,A,中去掉2,5两行后所得到的子矩阵,解法一：,a,=1,3,4；,B=A,(,a,:),解法二：,B=A(1,:);A(3,:);A(4,:),3. 矩阵的加减法、乘法、转置与求逆运算,A+B，A-B，A*B，A.2，A,inv(A),det(A,),分别表示：,A,B,的和,差,积,点乘方,转置,求逆,以及,A,的行列式,B=,1 5 1 0 1,3 7 1 0 1,4 8 0 1 1,例5. 已知,求：,AB，B,-1,，B-A,T,，|A|,解：,A=1,2,3;4,5,6;7,8,0;B=1,2,1;1,1,2;2,1,1;,a=A*B,b=inv(B),c=B-A,d=,det(A,),a = 9 7 8,21 19 20,15 22 23,b = -1/4 1/4 -3/4,3/4 -1/4 -1/4,-1/4 3/4 -1/4,c = 0 -2 -6,-1 -4 -6,-1 -5 1,det(A,)=27,4. 向量的数量积，矢量积与范数,的数量积,例如：,a=,1,2,3,b,=-1,5,6,c=1,0,1,则,Matlab,中数量积:,dot(,a,b,);,矢量积:,cross(,a,b,),dot,(,a,b,)=27,cross,(,a,c,)=(2,2,-2),解：,a,b,c,的混合积为：,dot,(,a,cross,(,b,c,),练习：计算,a,b,c,的混合积,Matlab,中向量,a,的范数为：,norm,(,a,),例6,a=,1,2,3,b,=-1,5,6,c=1,0,1,求,a,b,的范数,解：,norm,(,a,)= 3.7417 , norm(,b,)=7.8740,练习：对例6计算：,a,b,夹角的余弦,dot(a/norm(a),b/norm(b),解法一：,dot(a,b)/norm(a)/norm(b),解法二：,=0.9164,思考：,a,b,c,三个向量那两个更接近？,二,.,度量空间与距离,1.,度量空间,定义：设,X,是任一集合，如果,X,中任意两个元素,x,与,y,，都对应一个实数,d,(,x,y,),，且满足：,(1),非负性,:,d,(,x,y,)0,当且仅当,x,=,y,时,d,(,x,y,)=0,(2),对称性,:,d,(,x,y,)=,d,(,y, x,),(3),三角不等式,:,对任意的,x,y,z,X,，有,d,(,x,y,),d,(,x,z,)+,d,(,z,y,),则称,d,(,x,y,),为,x,与,y,之间的距离，称,X,是以,d,(,x,y,),为距离的度量空间或距离空间,.,2.,常见的距离空间,n,维欧氏空间,:,设 表示,n,维向量,的全体所组成的集合，称为,n,维欧氏空间,(2),连续函数空间,C,a,b,:,令,C,a,b,=,x,(,t,)|,x,(,t,),是,a,b,上的连续函数,则称,C,a,b,为连续函数空间，在,C,a,b,上定义,d,(,x,y,),=,max|,x,(,t,)-,y,(,t,),|,(3),平方可积函数空间 ： 令,则称 为平方可积函数空间 ，定义距离：,(4),平方可和离散序列空间 ：令,则称 平方可和离散序列空间，定义距离：,3.,向量的距离与计算,(1)欧氏距离,：,Matlab,中命令：,dist(A,B),计算,A,中每个行向量与,B,中每个列向量之间欧氏距离，,A,的行数必须等于,B,的列数.,例7.,a=,1,2,3,b,=-1,5,6,c=1,0,1,求,a,b,c,欧氏距离,解：,dist(,a,b,)=,4.6904,dist(,a,c,)=,2.8284,dist(c,b,)=,7.3485,(2)绝对距离：,Matlab,中命令：,mandist(A,B,),计算,A,中每个行向量与,B,中每个列向量之间绝对距离，,A,的行数必须等于,B,的列数.,例7. 求例6中向量之间的绝对值距离.,mandist(a,c,)=4；mandist(a,b)=8；,mandist(c,b,)=12,解,：,dist(,a,b,)=,4.6904,dist(,a,c,)=,2.8284,dist(c,b,)=,7.3485,由此，你发现了什么？,(3) 闵可夫斯基距离：,当,r=1,2,时分别为绝对距离和欧氏距离,(4) 马氏距离：,其中,V,是一个实对称正定矩阵，通常取样本的协方差矩阵，当,V=E,时即为欧氏距离.,以上距离，在,Matlab,(6.),中有命令:,pdist,设样本,X,是,m,个,n,维向量所组成的矩阵，则有：,Pdist(,X,) ,样本,X,中各,n,维向量的欧氏距离,Pdist(,X,cityblock,) ,各,n,维向量的绝对距离,Pdist(,X,Minkowski，r,) ,闵可夫斯基距离,Pdist(,X,mahal,) ,各,n,维向量的马氏距离,注意： 而,pdist(X,),是一行 列,各列表示,X,中各行向量按如下顺序的距离 (1,2),(1,3),(1,m),(2,3),(2,4),(2,m),(m-1,m),三,.,向量的均值、方差、协方差与相关矩阵,mean(A) ,A,中各,列向量的均值,Var(A,),A,中各,列向量的方差,Std(A),A,中各,列向量的标准差,Cov(A,),A,中各,列向量的协方差矩阵,Corrcoef(A,),A,中各,列向量的相关矩阵,如果计算,A,中各行向量的均值、方差、协方差矩阵，相关矩阵，只需先将,A,转置即可.,例8.,现测得6只,Apf,和9只,Af,蠓虫的触长,翅长数据如下：,Apf：(1.14,1.78), (1.18,1.96), (1.20,1.86), (1.26,2.00), (1.28,2.00), (1.30,1.96) Af：(1.24,1.72), (1.36,1.74), (1.38,1.64), (1.38,1.82), (1.38,1.90), (1.40,1.70), (1.48,1.82),(1.54,1.82), (1.56,2.08),计算两类蠓虫的各自之间的欧氏、绝对、马氏距离,解:,Af,=1.24,1.72;1.36,1.74;1.38,1.64;1.38,1.82; 1.38,1.90 ;,1.40,1.70;1.48,1.82;1.54,1.82;1.56,2.08；,d1=(,pdist(Apf,);,d2=(,pdist(Apf,cityblock,);,d3=,pdist(Apf,mahal,);,d=d1,d2,d3,Apf,=1.14,1.78;1.18,1.96;1.2,1.86;1.26,2.;1.28,2;,1.30,1.96 ;,表一.,Apf,蠓虫之间的距离,Apf,蠓虫,欧氏距离,绝对距离,马氏距离,d12,0.1844,0.2200,2.5626,d13,0.1000,0.1400,0.9883,d14,0.2506,0.3400,2.4942,d15,0.2608,0.3600,2.5318,d16,0.2408,0.3400,2.5478,d23,0.1020,0.1200,2.2507,d24,0.0894,0.1200,1.5470,d25,0.1077,0.1400,2.0430,d26,0.1200,0.1200,3.0777,d34,0.1523,0.2000,1.6534,d35,0.1612,0.2200,1.5873,d36,0.1414,0.2000,1.6025,d45,0.0200,0.0200,0.5129,d46,0.0566,0.0800,1.6616,d56,0.0447,0.0600,1.1764,Af,蠓,欧氏距,绝对距,马氏距,Af,蠓,欧氏距,绝对距,马氏距,d12,0.1217,0.1400,1.4423,d37,0.2059,0.2800,1.3971,d13,0.1612,0.2200,2.3963,d38,0.2408,0.3400,1.6847,d14,0.1720,0.2400,1.4225,d39,0.4754,0.6200,3.4103,d15,0.2280,0.3200,1.5517,d45,0.0800,0.0800,0.7917,d16,0.1612,0.1800,2.2078,d46,0.1217,0.1400,1.3659,d17,0.2600,0.3400,2.6110,d47,0.1000,0.1000,1.2987,d18,0.3162,0.4000,3.3635,d48,0.1600,0.1600,2.0780,d19,0.4817,0.6800,3.3694,d49,0.3162,0.4400,2.1271,d23,0.1020,0.1200,1.1705,d56,0.2010,0.2200,2.1520,d24,0.0825,0.1000,0.6601,d57,0.1281,0.1800,1.8990,d25,0.1612,0.1800,1.4345,d58,0.1789,0.2400,2.6482,d26,0.0566,0.0800,0.8277,d59,0.2546,0.3600,1.8449,d27,0.1442,0.2000,1.2266,d67,0.1442,0.2000,0.9689,d28,0.1970,0.2600,1.9404,d68,0.1844,0.2600,1.4149,d29,0.3945,0.5400,2.6612,d69,0.4123,0.5400,2.9389,d34,0.1800,0.1800,1.7814,d78,0.0600,0.0600,0.7792,d35,0.2600,0.2600,2.5731,d79,0.2720,0.3400,2.0832,d36,0.0632,0.0800,0.4756,d89,0.2608,0.2800,2.4183,如果我们要将这些距离按大小排序，如何快捷地实现？在,Matlab,中方法如下：,F=sort(d);F,i=sort(d),其中,sort(d),将,d,中距离按从小到大排列，而,i,中各列则指出了排序后各距离原来的序号.,i = 13 13 13,15 15 2,14 14 15,7 6 7,2 7 11,6 9 12,8 2 10,9 8 14,12 10 8,10 12 6,11 1 3,1 11 4,5 3 5,3 5 1,4 4 9,作业：,1.,（,1,）输入,A,B,并计算各自行列式的值，逆矩阵；,（,2,）,A,的行向量的协方差矩阵，,B,的列向量 的相关系数矩阵,（,3,）计算,A,列向量间各种距离，夹角余弦，那两个向量更接近？,（,4,）将矩阵,A,的各行向量标准化；并给出验证方法；,（,5,）求去掉,B,的第一行与第三列后的矩阵,C,，求,AC,C2,A.2,2.,将下表给出的数据粘帖到,matlab,，然后进行各指标排序，从而得到各省、市、自治区的各指标排名矩阵,D,安徽省排名如何？,地 区,企业单位数,工 业,工 业,实收资本,资 产,流动资产,流动资产,固定资产,(,个,),总 产 值,增 加 值,合 计,合 计,年平均余额,原价合计,全 国,31423,27220.91,7128.11,10256.78,28354.46,14029.75,13750.96,16700.30,北 京,1023,1318.03,295.54,410.09,1166.26,715.76,720.43,503.53,天 津,1350,1365.99,324.44,553.06,1421.84,746.63,714.29,759.44,河 北,760,424.94,122.38,223.92,625.90,249.95,234.68,411.51,山 西,101,74.70,25.81,71.94,207.67,49.90,44.99,141.82,内蒙古,89,65.92,20.12,29.83,122.25,80.91,78.02,44.65,辽 宁,1070,894.67,238.90,406.79,1097.95,462.38,447.83,822.40,吉 林,245,412.72,121.57,118.49,353.28,169.53,145.29,225.09,黑龙江,147,114.82,36.67,60.26,219.85,103.00,102.17,118.40,上 海,3669,3904.80,1024.35,1655.58,4385.45,2332.52,2324.98,2331.17,江 苏,3601,3333.31,835.68,1271.05,3311.92,1575.00,1557.56,2008.05,浙 江,2715,1532.12,364.72,503.43,1424.40,747.77,723.92,794.17,安 徽,277,246.80,78.12,149.86,364.69,141.13,138.21,251.70,福 建,2923,1822.48,490.68,689.96,1940.28,894.00,854.22,1179.72,江 西,177,106.87,29.42,46.50,139.40,63.58,64.86,79.85,山 东,2020,1457.01,431.75,493.52,1397.82,657.56,621.88,805.65,河 南,421,262.13,86.71,117.83,415.29,199.36,192.49,303.22,湖 北,392,382.38,135.04,196.03,529.20,214.44,191.99,311.53,湖 南,239,147.15,43.05,72.96,188.77,79.84,78.04,137.53,广 东,9020,8490.96,2153.18,2658.90,7679.21,3940.62,3932.87,4614.89,广 西,238,147.43,44.93,87.73,248.77,94.42,91.86,162.90,海 南,94,40.54,10.25,29.47,60.12,26.75,28.06,34.33,重 庆,151,179.87,56.44,119.17,260.17,110.92,112.61,208.34,四 川,268,188.83,60.12,109.31,287.94,127.31,117.23,156.47,贵 州,76,21.56,6.59,23.09,52.05,27.52,25.01,20.75,云 南,120,69.59,22.34,55.06,125.82,53.78,53.35,82.71,西 藏,5,0.04,0.02,0.33,0.50,0.20,0.18,0.28,陕 西,130,140.73,47.32,67.25,208.45,105.35,95.82,123.96,甘 肃,37,36.02,10.29,13.39,55.57,27.39,27.38,29.02,青 海,9,5.39,1.13,2.71,8.75,4.23,2.65,4.84,宁 夏,24,19.36,6.48,10.30,27.59,14.90,15.20,15.12,新 疆,32,13.74,4.06,8.99,27.31,13.13,12.91,17.22,