matlab第四章矩阵与方程组.ppt

第四讲矩阵与线性代数计算,研究内容：线性方程组的求解与矩阵特征值问题,一、矩阵定义,二、矩阵生成生成规则生成方式：由元素列表直接生成矩阵。从外部数据文件读入矩阵。用户自编M文件生成矩阵。利用小阵生成大阵。利用系统内部函数生成矩阵。,A=123;456;789;B=A,A+0.1;A+0.2,A+0.3B=1.00002.00003.00001.10002.10003.10004.00005.00006.00004.10005.10006.10007.00008.00009.00007.10008.10009.10001.20002.20003.20001.30002.30003.30004.20005.20006.20004.30005.30006.30007.20008.20009.20007.30008.30009.3000,（1）zeros：生成元素全部为0的矩阵。B=zeros(3,4)B=000000000000（2）ones：生成元素全部为1的矩阵。C=ones(2,5)C=1111111111（3）rand：生成均匀分布随机元素矩阵。D=rand(3,5)D=0.95010.48600.45650.44470.92180.23110.89130.01850.61540.73820.60680.76210.82140.79190.1763,（4）randn：生成正态分布随机元素矩阵。E=randn(2,4)E=-0.43260.1253-1.14651.1892-1.66560.28771.1909-0.0376（5）magic：生成N阶幻方方阵。M=magic(4)M=16231351110897612414151（6）diag：生成一个矩阵的主对角元素。Mo=diag(M)Mo=161161,diag(Mo)ans=160000110000600001（7）triu：生成上三角阵。M1=triu(M)M1=162313011108006120001（8）tril：生成下三角阵。M2=16000M2=tril(M)511009760414151,（9）length：用来返回指定向量的长度。n=length(Mo)n=4（10）size：返回指定矩阵的行数和列数。m,n=size(M)m=4n=4（11）eye：生成指定行列数的单位矩阵。e=eye(size(M)e=1000010000100001（12）hilb：生成以1/i+j-1为元素的实对称矩阵。,h4=hilb(4)h4=1.00000.50000.33330.25000.50000.33330.25000.20000.33330.25000.20000.16670.25000.20000.16670.1429三、矩阵取块,A=1.10001.20001.30001.40002.10002.20002.30002.40003.10003.20003.30003.4000A(2,:)ans=2.10002.20002.30002.4000A(:,3)ans=1.30002.30003.3000A(2:7)ans=2.10003.10001.20002.20003.20001.3000,A(:,2:4)ans=1.20001.30001.40002.20002.30002.40003.20003.30003.4000A(:)ans=1.10002.10003.10001.20002.20003.20001.30002.30003.30001.40002.40003.4000,四、矩阵运算,矩阵的行列式与转置阵行列式：det(A)A1=123;456;780;A10=det(A1)A10=27A2=123;456;789;A20=det(A2)A20=0symscA3=123;456;78c;A30=det(A3)A30=-3*c+27,转置矩阵：AB1=135;2610;BB=B1,AA=A1BB=1236510AA=147258360矩阵的加、减、乘及乘方运算注：与数学中的运算符大致一样，可开任意有理数根。矩阵求逆与除法运算(满秩方阵)A-1=inv(A),A/B=A*inv(B),AB=inv(A)*B,inv(A1)ans=-1.77780.8889-0.11111.5556-0.77780.2222-0.11110.2222-0.1111inv(A3)ans=-1/3*(5*c-48)/(c-9),2/3*(c-12)/(c-9),1/(c-9)2/3*(2*c-21)/(c-9),-1/3*(c-21)/(c-9),-2/(c-9)1/(c-9),-2/(c-9),1/(c-9)例1：求解方程组AX=b,A=11；11，b=5;1A=1-1;11;b=5;1;X=inv(A)*b;或X=Abinv(A),Xans=0.50000.5000X=3-0.50000.5000-2,矩阵秩与奇异值rank(A)A1=123;456;780;A2=123;456;789;B1=135;2610;rank(A1)ans=3rank(A2)ans=2rank(B1)ans=1奇异值,矩阵范数与条件数范数：norm(A,p)b=123;A12=norm(A1,2);A22=norm(A2,2);b2=norm(b,2);A12,A22,b2A12=13.2015A22=16.8481b2=3.7417条件数：cond(A)K1=cond(A1)K1=35.1059K2=cond(A2)K2=3.8131e+016病态方程组,五、矩阵特征值和特征向量,P,R=eig(A)实对称矩阵：A1=5-20;-262;027;P,R=eig(A1)P=0.66670.6667-0.33330.6667-0.33330.6667-0.33330.66670.6667R=3.00000006.00000009.0000,实对称矩阵：A2=1-1-1;-11-1;-1-11;P,R=eig(A2)P=0.57740.76340.28950.5774-0.63250.51640.5774-0.1310-0.8059R=-1.00000002.00000002.0000,B2=122;1-11;4-121;P,R=eig(B2)P=0.9045-0.7255-0.72550.3015-0.2176-0.0725i-0.2176+0.0725i-0.30150.5804-0.2902i0.5804+0.2902iR=1.0000000-0.0000+1.0000i000-0.0000-1.0000i,六、矩阵分解与广义逆阵,广义逆阵：设A为m*n阶矩阵，如果存在n*m阶矩阵G,满足条件(1)AGA=A,(2)GAG=G,(3)(AG)*=AG,(4)(GA)*=GA,*表示共轭后再转置，则称G为A的广义逆阵，记为A+=G。A+=pinv(A)A1=123;456;780;G1=pinv(A1)G1=-1.77780.8889-0.11111.5556-0.77780.2222-0.11110.2222-0.1111,A2=123;456;G2=pinv(A2)G2=-0.94440.4444-0.11110.11110.7222-0.2222A2*G2ans=1.00000-0.00001.0000G2*A2ans=0.83330.3333-0.16670.33330.33330.3333-0.16670.33330.8333,A3=12;24;36;G3=pinv(A3)G3=0.01430.02860.04290.02860.05710.0857A3*G3ans=0.07140.14290.21430.14290.28570.42860.21430.42860.6429G3*A3ans=0.20000.40000.40000.8000,LU分解A=LU，A是方阵，U是上三角阵，L是一个经过行交换的下三角阵。L,U=lu(A)A1=5-20;-262;027;L1,U1=lu(A1)L1=1.000000-0.40001.0000000.38461.0000U1=5.0000-2.0000005.20002.0000006.2308,Cholesky分解A=RR,A为对称正定矩阵，R为上三角阵，R为R的转置。R=chol(A)A=5-10;-262;027;R=chol(A)R=2.2361-0.4472002.40830.8305002.5120,QR分解A=QR,Q为正交阵，R为上三角阵。Q,R=qr(A)A1=5-20;-262;027;Q1,R1=qr(A1)Q1=-0.9285-0.3431-0.14210.3714-0.8578-0.35530-0.38270.9239R1=-5.38524.08530.74280-5.2259-4.3945005.7564,奇异值分解A=U*S*V*,S为A的奇异值对角阵，U与V为正交阵，V*是V的共轭后再转置。U,S,V=svd(A)A1=123;234;345;U1,S1,V1=svd(A1)U1=-0.38510.82770.4082-0.55950.1424-0.8165-0.7339-0.54280.4082S1=9.62350000.62350000.0000V1=-0.3851-0.82770.4082-0.5595-0.1424-0.8165-0.73390.54280.4082,例2：求解线性方程组x1+2x2+3x3=62x1+3x2+4x3=93x1+4x2+7x3=14解：A1=123;234;347;b1=6;9;14;r1=rank(A1)r1=3所以A1为一满秩方阵，方程组有惟一解。X11=A1b1X11=1.00001.00001.0000,七、线性方程组求解,例3：求解x1+2x2+3x3=62x1+3x2+4x3=93x1+5x2+7x3=15解：A2=123;234;357;b2=6;9;15;r12=rank(A2)r12=2Ab2=A2,b2Ab2=1236234935715r22=rank(Ab2)r22=2由r22=r21=2知，方程组为相容方程组，只有两个独立方程，可取x1+2x2+3x3=62x1+3x2+4x3=9,若令x3=c,利用线性方程组的符号解法有：A20=12;23;b20=sym(6-3*c;9-4*c);X20=A20b20X20=c3-2*c解为X=x1,x2,x3=c,3-2c,c,例4：求解x1+2x2+3x3=62x1+4x2+6x3=123x1+6x2+9x3=18解：A3=123;246;369;b3=6;12;18;r13=rank(A3)r13=1Ab3=A3,b3;r32=rank(Ab3)r32=1由r31=r32=1知方程为相容方程组，且只有一个方程独立，可取x1+2x2+3x3=6。若令x1=c1,x2=c2,则x3=2-1/3c1-2/3c2。解为x1=c1,x2=c2,x3=2-1/3c1-2/3c2。,例5：求解x1+2x2+3x3=62x1+3x2+4x3=93x1+5x2+7x3=14解：A4=123;234;357;b4=6;9;14;r41=rank(A4)r41=2Ab4=A4,b4;r42=rank(Ab4)r42=3由r41=r42知方程组为矛盾方程组，不存在通常意义下的解，但可利用广义逆阵求最小二乘解，X=pinv(A4)*b4X=1.16671.00000.8333,