matlab answer.docx

1、matlab answer%Chapter 3 %Exercise 1 a=1,2,3;b=2,4,3;a./b,a.b,a/b,abans = 0.5000 0.5000 1.0000ans = 2 2 1ans = 0.6552 %一元方程组x2,4,3=1,2,3的近似解ans = 0 0 0 0 0 0 0.6667 1.3333 1.0000%矩阵方程1,2,3x11,x12,x13;x21,x22,x23;x31,x32,x33=2,4,3的特解 %Exercise 2(1) A=4 1 -1;3 2 -6;1 -5 3;b=9;-2;1; rank(A), rank(A,b) %

2、A,b为增广矩阵ans = 3ans = 3 %可见方程组唯一解 x=Abx = 2.3830 1.4894 2.0213 %Exercise 2(2) A=4 -3 3;3 2 -6;1 -5 3;b=-1;-2;1; rank(A), rank(A,b)ans = 3ans = 3 %可见方程组唯一解 x=Abx = -0.4706 -0.2941 0 %Exercise 2(3) A=4 1;3 2;1 -5;b=1;1;1; rank(A), rank(A,b)ans = 2ans = 3 %可见方程组无解 x=Abx = 0.3311 -0.1219 %最小二乘近似解 %Exerci

3、se 2(4) a=2,1,-1,1;1,2,1,-1;1,1,2,1;b=1 2 3;%注意b的写法 rank(a),rank(a,b)ans = 3ans = 3 %rank(a)=rank(a,b) abans = 1 0 1 0 %一个特解 %Exercise 3 a=2,1,-1,1;1,2,1,-1;1,1,2,1;b=1,2,3; ksi=null(a),x0=abksi = -0.6255 0.6255 -0.2085 0.4170x0 = 1 0 1 0%通解x=k*ksi+x0另一种解法 rref(a,b)ans = 1.0000 0 0 1.5000 1.0000 0 1

4、.0000 0 -1.5000 0 0 0 1.0000 0.5000 1.0000%前4列分别对应x1-x4, 通解x=k*-1.5,1.5,-0.5,1+1,0,1,0 %Exercise 4%一年后城市=上年城市99%+上年农村5%；%一年后农村=上年城市1%+上年农村95% x0=0.2 0.8;a=0.99 0.05;0.01 0.95; x1=a*x0, x2=a2*x0, x10=a10*x0 x=x0;for i=1:1000,x=a*x;end,xx = 0.8333 0.1667 x0=0.7 0.3; x=x0;for i=1:1000,x=a*x;end,xx = 0.

5、8333 0.1667 %注意结果不受x0改变的影响 v,e=eig(a)v = 0.9806 -0.7071 0.1961 0.7071e = 1.0000 0 0 0.9400 v(:,1)./xans = 1.1767 1.1767 %v(:,1)与x成比例，说明x也是最大特征值1对应的特征向量 %Exercise 5%用到公式(3.11)(3.12) B=6,2,1;2.25,1,0.2;3,0.2,1.8;x=25 5 20; C=B/diag(x) %第1列除以x(1),以此类推C = 0.2400 0.4000 0.0500 0.0900 0.2000 0.0100 0.1200

6、 0.0400 0.0900 A=eye(3,3)-CA = 0.7600 -0.4000 -0.0500 -0.0900 0.8000 -0.0100 -0.1200 -0.0400 0.9100 D=17 17 17;x=AD %D变了，但假设C,A不变x = 37.5696 25.7862 24.7690 %Exercise 6(1) a=4 1 -1;3 2 -6;1 -5 3;det(a),inv(a),v,d=eig(a)ans = -94ans = 0.2553 -0.0213 0.0426 0.1596 -0.1383 -0.2234 0.1809 -0.2234 -0.053

7、2v = 0.0185 -0.9009 -0.3066 -0.7693 -0.1240 -0.7248 -0.6386 -0.4158 0.6170d = -3.0527 0 0 0 3.6760 0 0 0 8.3766 %Exercise 6(2) a=1 1 -1;0 2 -1;-1 2 0;det(a),inv(a),v,d=eig(a)ans = 1ans = 2.0000 -2.0000 1.0000 1.0000 -1.0000 1.0000 2.0000 -3.0000 2.0000v = -0.5773 0.5774 + 0.0000i 0.5774 - 0.0000i -0

8、.5773 0.5774 0.5774 -0.5774 0.5773 - 0.0000i 0.5773 + 0.0000id = 1.0000 0 0 0 1.0000 + 0.0000i 0 0 0 1.0000 - 0.0000i %Exercise 6(3) A=5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10A = 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 det(A),inv(A), v,d=eig(A)ans = 1ans = 68.0000 -41.0000 -17.0000 10.0000 -41.0000 25.0000 10.

9、0000 -6.0000 -17.0000 10.0000 5.0000 -3.0000 10.0000 -6.0000 -3.0000 2.0000v = 0.8304 0.0933 0.3963 0.3803 -0.5016 -0.3017 0.6149 0.5286 -0.2086 0.7603 -0.2716 0.5520 0.1237 -0.5676 -0.6254 0.5209d = 0.0102 0 0 0 0 0.8431 0 0 0 0 3.8581 0 0 0 0 30.2887 %Exercise 6(4)(以n=5为例，n=50或500类似) %关键是矩阵的定义，当n很

10、大时不能直接写出。%方法一（三个for）n=5;for i=1:n, a(i,i)=5;endfor i=1:(n-1),a(i,i+1)=6;endfor i=1:(n-1),a(i+1,i)=1;enda%方法二（一个for）n=5;a=zeros(n,n);a(1,1:2)=5 6;for i=2:(n-1),a(i,i-1,i,i+1)=1 5 6;enda(n,n-1 n)=1 5;a%方法三（不用for）n=5;a=diag(5*ones(n,1);b=diag(6*ones(n-1,1);c=diag(ones(n-1,1);a=a+zeros(n-1,1),b;zeros(1,

11、n)+zeros(1,n);c,zeros(n-1,1)%下列计算 det(a)ans = 665 inv(a)ans = 0.3173 -0.5865 1.0286 -1.6241 1.9489 -0.0977 0.4887 -0.8571 1.3534 -1.6241 0.0286 -0.1429 0.5429 -0.8571 1.0286 -0.0075 0.0376 -0.1429 0.4887 -0.5865 0.0015 -0.0075 0.0286 -0.0977 0.3173 v,d=eig(a)v = -0.7843 -0.7843 -0.9237 0.9860 -0.923

12、7 0.5546 -0.5546 -0.3771 -0.0000 0.3771 -0.2614 -0.2614 0.0000 -0.1643 0.0000 0.0924 -0.0924 0.0628 -0.0000 -0.0628 -0.0218 -0.0218 0.0257 0.0274 0.0257d = 0.7574 0 0 0 0 0 9.2426 0 0 0 0 0 7.4495 0 0 0 0 0 5.0000 0 0 0 0 0 2.5505 %Exercise 7(1) a=4 1 -1;3 2 -6;1 -5 3;v,d=eig(a)v = 0.0185 -0.9009 -0

13、.3066 -0.7693 -0.1240 -0.7248 -0.6386 -0.4158 0.6170d = -3.0527 0 0 0 3.6760 0 0 0 8.3766 det(v)ans = -0.9255 %v行列式正常, 特征向量线性相关，可对角化 inv(v)*a*v %验算ans = -3.0527 0.0000 -0.0000 0.0000 3.6760 -0.0000 -0.0000 -0.0000 8.3766 v2,d2=jordan(a) %另一方法：用jordanv2 = 0.0798 0.0076 0.9127 0.1886 -0.3141 0.1256 -0

14、.1605 -0.2607 0.4213 %用jordan特征向量不同d2 = 8.3766 0 0 0 -3.0527 - 0.0000i 0 0 0 3.6760 + 0.0000i v2a*v2ans = 8.3766 0 0.0000 0.0000 -3.0527 0.0000 0.0000 0.0000 3.6760 v(:,1)./v2(:,2) %对应相同特征值的特征向量成比例，说明两种方法结果本质上相同ans = 2.4491 2.4491 2.4491 %Exercise 7(2) a=1 1 -1;0 2 -1;-1 2 0;v,d=eig(a)v = -0.5773 0.

15、5774 + 0.0000i 0.5774 - 0.0000i -0.5773 0.5774 0.5774 -0.5774 0.5773 - 0.0000i 0.5773 + 0.0000id = 1.0000 0 0 0 1.0000 + 0.0000i 0 0 0 1.0000 - 0.0000i det(v)ans = -5.0566e-028 -5.1918e-017i %v的行列式接近0, 特征向量线性相关，不可对角化 v,d=jordan(a)v = 1 0 1 1 0 0 1 -1 0d = 1 1 0 0 1 1 0 0 1 %jordan标准形不是对角的，所以不可对角化 %E

16、xercise 7(3) A=5 7 6 5;7 10 8 7;6 8 10 9;5 7 9 10A = 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 v,d=eig(A)v = 0.8304 0.0933 0.3963 0.3803 -0.5016 -0.3017 0.6149 0.5286 -0.2086 0.7603 -0.2716 0.5520 0.1237 -0.5676 -0.6254 0.5209d = 0.0102 0 0 0 0 0.8431 0 0 0 0 3.8581 0 0 0 0 30.2887 inv(v)*A*vans = 0.0102 0

17、.0000 -0.0000 0.0000 0.0000 0.8431 -0.0000 -0.0000 -0.0000 0.0000 3.8581 -0.0000 -0.0000 -0.0000 0 30.2887%但本题用jordan不行, 原因未知 %Exercise 7(4)参考6(4)和7(1), 略 %Exercise 8 只有(3)对称且特征值全部大于零, 所以只有(3)是正定矩阵. %Exercise 9 a=4 -3 1 3;2 -1 3 5;1 -1 -1 -1;3 -2 3 4;7 -6 -7 0 rank(a)ans = 3 rank(a(1:3,:)ans = 2 ran

18、k(a(1 2 4,:) %1,2,4行为最大无关组ans = 3 b=a(1 2 4,:);c=a(3 5,:); bc %线性表示的系数ans = 0.5000 5.0000 -0.5000 1.0000 0 -5.0000 %Exercise 10 a=1 -2 2;-2 -2 4;2 4 -2 v,d=eig(a)v = 0.3333 0.9339 -0.1293 0.6667 -0.3304 -0.6681 -0.6667 0.1365 -0.7327d = -7.0000 0 0 0 2.0000 0 0 0 2.0000 v*vans = 1.0000 0.0000 0.0000

19、 0.0000 1.0000 0 0.0000 0 1.0000 %可见v确实是正交矩阵%正交变换为y=vx, 标准型-7*y12+2*y22+2*y32 %Exercise 11%设经过6个电阻的电流分别为i1, ., i6. 列方程组如下%20-2i1=a; 5-3i2=c; a-3i3=c; a-4i4=b; c-5i5=b; b-3i6=0;%i1=i3+i4;i5=i2+i3;i6=i4+i5;%计算如下 A=1 0 0 2 0 0 0 0 0;0 0 1 0 3 0 0 0 0;1 0 -1 0 0 -3 0 0 0; 1 -1 0 0 0 0 -4 0 0;0 -1 1 0 0

20、0 0 -5 0;0 1 0 0 0 0 0 0 -3; 0 0 0 1 0 -1 -1 0 0;0 0 0 0 -1 -1 0 1 0;0 0 0 0 0 0 -1 -1 1; b=20 5 0 0 0 0 0 0 0; Abans = 13.3453 6.4401 8.5420 3.3274 -1.1807 1.6011 1.7263 0.4204 2.1467 %Exercise 12 A=1 2 3;4 5 6;7 8 0; left=sum(eig(A), right=sum(trace(A)left = 6.0000right = 6 left=prod(eig(A), right=det(A) %原题有错, (-1)n应删去left = 27.0000right = 27 fA=(A-p(1)*eye(3,3)*(A-p(2)*eye(3,3)*(A-p(3)*eye(3,3)fA = 1.0e-012 * 0.0853 0.1421 0.0284 0.1421 0.1421 0 -0.0568 -0.1137 0.1705 norm(fA) ans = 2.9536e-013 %f(A)范数接近0，也就是说fA是0矩阵。