大连理工大学线性代数实验上机报告.docx

大连理工大学线性代数实验上机报告实验一首先随机生成五阶方阵AA=rand(5)A = 0.8147 0.0975 0.1576 0.1419 0.6557 0.9058 0.2785 0.9706 0.4218 0.0357 0.1270 0.5469 0.9572 0.9157 0.8491 0.9134 0.9575 0.4854 0.7922 0.9340 0.6324 0.9649 0.8003 0.9595 0.6787 B=rand(5)随机生成五阶方阵BB = 0.7577 0.7060 0.8235 0.4387 0.4898 0.7431 0.0318 0.6948 0.3816 0.4456 0.3922 0.2769 0.3171 0.7655 0.6463 0.6555 0.0462 0.9502 0.7952 0.7094 0.1712 0.0971 0.0344 0.1869 0.7547 b=rand(1,5)随机生成列向量bb = 0.2760 0.6797 0.6551 0.1626 0.1190计算A+B A+Bans = 1.5725 0.8036 0.9811 0.5806 1.1455 1.6489 0.3103 1.6654 0.8033 0.4813 0.5192 0.8238 1.2743 1.6813 1.4954 1.5689 1.0037 1.4356 1.5874 1.6434 0.8035 1.0620 0.8347 1.1464 1.4334计算A-B A-Bans = 0.0570 -0.6085 -0.6658 -0.2969 0.1660 0.1627 0.2467 0.2758 0.0402 -0.4099 -0.2652 0.2700 0.6401 0.1502 0.2028 0.2579 0.9113 -0.4648 -0.0030 0.2246 0.4612 0.8678 0.7658 0.7726 -0.0760计算A*B+B*A A*B+B*Aans = 3.0288 2.3058 3.1439 2.7276 3.1034 2.9094 2.1967 3.0040 3.0737 3.2584 3.3422 2.1423 3.2104 3.5734 3.9049 4.1446 2.9794 4.3676 4.2354 4.9170 3.1350 1.7787 3.2289 3.1170 3.2815求Ax=b的解 x=Abx = -0.9850 2.4396 3.3124 -5.6515 1.7085验证克莱姆法则 c=A(:,1)c = 0.8147 0.9058 0.1270 0.9134 0.6324 d=A(:,2)d = 0.0975 0.2785 0.5469 0.9575 0.9649 e=A(:,3)e = 0.1576 0.9706 0.9572 0.4854 0.8003 f=A(:,4)f = 0.1419 0.4218 0.9157 0.7922 0.9595 g=A(:,5)g = 0.6557 0.0357 0.8491 0.9340 0.6787 B1=b;d;e;f;gB1 = 0.2760 0.0975 0.1576 0.1419 0.6557 0.6797 0.2785 0.9706 0.4218 0.0357 0.6551 0.5469 0.9572 0.9157 0.8491 0.1626 0.9575 0.4854 0.7922 0.9340 0.1190 0.9649 0.8003 0.9595 0.6787 B2=c;b;e;f;gB2 = 0.8147 0.2760 0.1576 0.1419 0.6557 0.9058 0.6797 0.9706 0.4218 0.0357 0.1270 0.6551 0.9572 0.9157 0.8491 0.9134 0.1626 0.4854 0.7922 0.9340 0.6324 0.1190 0.8003 0.9595 0.6787 B3=c;d;b;f;gB3 = 0.8147 0.0975 0.2760 0.1419 0.6557 0.9058 0.2785 0.6797 0.4218 0.0357 0.1270 0.5469 0.6551 0.9157 0.8491 0.9134 0.9575 0.1626 0.7922 0.9340 0.6324 0.9649 0.1190 0.9595 0.6787 B4=c;d;e;b;gB4 = 0.8147 0.0975 0.1576 0.2760 0.6557 0.9058 0.2785 0.9706 0.6797 0.0357 0.1270 0.5469 0.9572 0.6551 0.8491 0.9134 0.9575 0.4854 0.1626 0.9340 0.6324 0.9649 0.8003 0.1190 0.6787 B5=c;d;e;f;bB5 = 0.8147 0.0975 0.1576 0.1419 0.2760 0.9058 0.2785 0.9706 0.4218 0.6797 0.1270 0.5469 0.9572 0.9157 0.6551 0.9134 0.9575 0.4854 0.7922 0.1626 0.6324 0.9649 0.8003 0.9595 0.1190 x1=det(B1)/det(A)x1 = -0.9850 x2=det(B2)/det(A)x2 = 2.4396 x3=det(B3)/det(A)x3 = 3.3124 x4=det(B4)/det(A)x4 = -5.6515 x5=det(B5)/det(A)x5 = 1.7085计算A的行列式 det(A)ans = -0.0250计算B的行列式 det(B)ans = 0.0647求A的逆 inv(A)ans = 3.1375 -0.8078 -1.8788 -4.2194 5.1680 -8.6076 3.5314 2.8907 13.7204 -14.3665 -6.2824 3.7220 3.6132 10.0084 -12.4190 13.6173 -6.8822 -6.3938 -23.5288 27.5825 -2.5292 1.0729 2.4193 5.8870 -7.2671求B的逆 inv(B)ans = -0.4430 3.4997 1.3255 -2.6005 -0.4697 1.4047 -1.1626 0.2422 -0.4475 -0.0119 0.7210 -1.8189 -2.0635 2.4434 0.0765 -0.6122 -0.1837 2.0165 0.0375 -1.2564 0.0384 -0.5157 -0.7370 0.5267 1.7407求A的秩 rank(A)ans = 5求B的秩 rank(B)ans = 5求A*B的行列式 det(A*B)ans = -0.0016求A*B的逆 inv(A*B)ans = -74.0649 35.0433 31.2288 121.5740 -137.3442 6.8291 -1.2718 -2.2922 -8.9951 8.6972 63.9620 -31.4202 -29.5061 -105.6918 122.3246 -9.3196 5.7452 4.6259 11.9660 -15.4028 11.9582 -6.3521 -3.3817 -16.7574 18.6360 rank(A*B)ans = 5 det(A)*det(B)ans = -0.0016验证（1） (A*B)ans = 0.9569 1.5566 1.6237 2.2732 2.2552 0.6922 0.9401 0.4969 0.9371 0.8090 0.9461 1.6492 1.6875 2.3563 2.3800 0.7507 1.5887 1.8840 1.9421 2.1481 1.1399 1.5212 2.2149 2.4545 2.4497 B*Aans = 0.9569 1.5566 1.6237 2.2732 2.2552 0.6922 0.9401 0.4969 0.9371 0.8090 0.9461 1.6492 1.6875 2.3563 2.3800 0.7507 1.5887 1.8840 1.9421 2.1481 1.1399 1.5212 2.2149 2.4545 2.4497（2） inv(B)*inv(A)ans = -74.0649 35.0433 31.2288 121.5740 -137.3442 6.8291 -1.2718 -2.2922 -8.9951 8.6972 63.9620 -31.4202 -29.5061 -105.6918 122.3246 -9.3196 5.7452 4.6259 11.9660 -15.4028 11.9582 -6.3521 -3.3817 -16.7574 18.6360（3） A*Bans = 0.9569 0.6922 0.9461 0.7507 1.1399 1.5566 0.9401 1.6492 1.5887 1.5212 1.6237 0.4969 1.6875 1.8840 2.2149 2.2732 0.9371 2.3563 1.9421 2.4545 2.2552 0.8090 2.3800 2.1481 2.4497 B*Aans = 2.0719 1.6135 2.1978 1.9769 1.9635 1.3528 1.2566 1.3549 1.4850 1.7372 1.7186 1.6454 1.5229 1.6894 1.6900 1.8714 2.0423 2.0113 2.2932 2.4625 0.8797 0.9697 0.8489 0.9690 0.8317求矩阵X使得AXB=C首先随机生成五阶方阵C C=rand(5)C = 0.4984 0.7513 0.9593 0.8407 0.3500 0.9597 0.2551 0.5472 0.2543 0.1966 0.3404 0.5060 0.1386 0.8143 0.2511 0.5853 0.6991 0.1493 0.2435 0.6160 0.2238 0.8909 0.2575 0.9293 0.4733X=A的逆*B的逆 X=inv(A)*C*inv(B)X = 3.8432 -13.8858 2.1418 9.4404 -4.5871 -9.3312 41.9602 -7.9101 -28.4683 14.8942 -7.8738 35.1218 -5.4107 -22.8861 10.1581 16.7545 -75.6079 14.6784 49.3951 -24.7450 -3.5568 17.0848 -2.9018 -11.2670 5.4559实验二1. 验证：对于一般的方阵A,B,C,D， 首先随机生成方阵A,B,C,DA=rand(5)A = 0.8258 0.1067 0.8687 0.4314 0.1361 0.5383 0.9619 0.0844 0.9106 0.8693 0.9961 0.0046 0.3998 0.1818 0.5797 0.0782 0.7749 0.2599 0.2638 0.5499 0.4427 0.8173 0.8001 0.1455 0.1450 B=rand(5)B = 0.8530 0.0760 0.4173 0.4893 0.7803 0.6221 0.2399 0.0497 0.3377 0.3897 0.3510 0.1233 0.9027 0.9001 0.2417 0.5132 0.1839 0.9448 0.3692 0.4039 0.4018 0.2400 0.4909 0.1112 0.0965 C=rand(5)C = 0.1320 0.2348 0.1690 0.5470 0.1835 0.9421 0.3532 0.6491 0.2963 0.3685 0.9561 0.8212 0.7317 0.7447 0.6256 0.5752 0.0154 0.6477 0.1890 0.7802 0.0598 0.0430 0.4509 0.6868 0.0811 D=rand(5)D = 0.9294 0.3063 0.6443 0.9390 0.2077 0.7757 0.5085 0.3786 0.8759 0.3012 0.4868 0.5108 0.8116 0.5502 0.4709 0.4359 0.8176 0.5328 0.6225 0.2305 0.4468 0.7948 0.3507 0.5870 0.8443 Z=A,B;C,DZ =0.8258 0.1067 0.8687 0.4314 0.13610.8530 0.0760 0.4173 0.4893 0.78030.5383 0.9619 0.0844 0.9106 0.86930.6221 0.2399 0.0497 0.3377 0.38970.9961 0.0046 0.3998 0.1818 0.5797 0.3510 0.1233 0.9027 0.9001 0.24170.0782 0.7749 0.2599 0.2638 0.5499 0.5132 0.1839 0.9448 0.3692 0.40390.4427 0.8173 0.8001 0.1455 0.1450 0.4018 0.2400 0.4909 0.1112 0.09650.1320 0.2348 0.1690 0.5470 0.1835 0.9294 0.3063 0.6443 0.9390 0.20770.9421 0.3532 0.6491 0.2963 0.3685 0.7757 0.5085 0.3786 0.8759 0.30120.9561 0.8212 0.7317 0.7447 0.6256 0.4868 0.5108 0.8116 0.5502 0.47090.5752 0.0154 0.6477 0.1890 0.7802 0.4359 0.8176 0.5328 0.6225 0.23050.0598 0.0430 0.4509 0.6868 0.0811 0.4468 0.7948 0.3507 0.5870 0.8443求Z的行列式 det(Z)ans = -0.0295求det(A)*det(D)-det(B)*det(C) det(A)*det(D)-det(B)*det(C)ans = 1.8656e-004随机生成对角矩阵A A=diag(rand rand rand rand rand)A = 0.1948 0 0 0 0 0 0.2259 0 0 0 0 0 0.1707 0 0 0 0 0 0.2277 0 0 0 0 0 0.4357随机生成对角矩阵B B=diag(rand rand rand rand rand)B = 0.3111 0 0 0 0 0 0.9234 0 0 0 0 0 0.4302 0 0 0 0 0 0.1848 0 0 0 0 0 0.9049随机生成对角矩阵C C=diag(rand rand rand rand rand)C = 0.9797 0 0 0 0 0 0.4389 0 0 0 0 0 0.1111 0 0 0 0 0 0.2581 0 0 0 0 0 0.4087随机生成对角矩阵D D=diag(rand rand rand rand rand)D = 0.5949 0 0 0 0 0 0.2622 0 0 0 0 0 0.6028 0 0 0 0 0 0.7112 0 0 0 0 0 0.2217 Z=A,B;C,DZ =0.1948 0 0 0 0 0.3111 0 0 0 0 0 0.2259 0 0 0 0 0.9234 0 0 0 0 0 0.1707 0 0 0 0 0.4302 0 0 0 0 0 0.2277 0 0 0 0 0.1848 0 0 0 0 0 0.4357 0 0 0 0 0.90490.9797 0 0 0 0 0.5949 0 0 0 0 0 0.4389 0 0 0 0 0.2622 0 0 0 0 0 0.1111 0 0 0 0 0.6028 0 00 0 0 0.2581 0 0 0 0 0.7112 00 0 0 0 0.4087 0 0 0 0 0.2217计算Z的行列式 det(Z)ans = -1.1243e-004计算det(A)*det(D)-det(B)*det(C) det(A)*det(D)-det(B)*det(C)ans = -9.3107e-005计算A*D-B*C的行列式 det(A*D-B*C)ans = -1.1243e-004实验三求A列向量组的一个最大无关组，并把不属于 极大无关组的向量利用极大无关组表示.N= 200865083;a=83;b=86;c=50;d=88;e=28;f=63;g=83;h=60; A=a,b,c,d,3,4;1,2,3,4,4,3;12,15,22,17,5,7;e,f,g,h,8,0; B=rref(A)B = 1.0000 0 0 0 -0.3548 0.4656 0 1.0000 0 0 -1.4905 -2.0020 0 0 1.0000 0 0.0473 0.3950 0 0 0 1.0000 1.7984 1.3383所以a1,a2,a3,a4是一个极大无关组。a5=-0,3548a1-1.4905a2+0.0473a3+1.7984a4;a6=0.4656a1-2.0020a2+0.3950a3+1.3383a4.实验四 a=83; b=86; c=50; d=88; e=28; f=63; g=83; h=60; b1=1,1.9,f,c; b2=1,1.8,f,c; A1=a,b,c,d;0.5,1,1.5,2;12,15,22,17;e,f,g,h; A2=a,b,c,d;0.3,0.6,0.9,1.2;12,15,22,17;e,f,g,h; A3=a,b,c,d;0.1,0.2,0.3,0.4;12,15,22,17;e,f,g,h; A4=a,b,c,d;0.05,0.1,0.15,0.2;12,15,22,17;e,f,g,h; x1=A1/b1x1 = 1.2043 0.0304 0.3517 1.2940 x2=A2/b1x2 = 1.2043 0.0182 0.3517 1.2940 x3=A3/b1x3 = 1.2043 0.0061 0.3517 1.2940 x4=A4/b1x4 = 1.2043 0.0030 0.3517 1.2940 x5=A2/b1x5 = 1.2043 0.0182 0.3517 1.2940 x6=A2/b2x6 = 1.2031 0.0182 0.3514 1.2931 x5=A1/b2x5 = 1.2031 0.0304 0.3514 1.2931 x6=A2/b2x6 = 1.2031 0.0182 0.3514 1.2931 x7=A3/b2x7 = 1.2031 0.0061 0.3514 1.2931 x8=A4/b2x8 = 1.2031 0.0030 0.35141.2931实验五随机生成4个5维向量，并进行正交化随机生成4个5维向量a=rand(4,5); aa = 0.1174 0.5079 0.0292 0.5785 0.5468 0.2967 0.0855 0.9289 0.2373 0.5211 0.3188 0.2625 0.7303 0.4588 0.2316 0.4242 0.8010 0.4886 0.9631 0.4889正交化 b=null(a)b = 0.6777 -0.5941 -0.3009 0.28920.1163实验六随机生成5阶矩阵 ，求其特征值及对应特征向量eig(A)ans = 3.2037 0.5684 -0.1026 + 0.5388i -0.1026 - 0.5388i -0.0457 d,v=eig(A)d = 0.2283 -0.5354 0.0179 + 0.1710i 0.0179 - 0.1710i -0.1383 0.3243 0.1499 0.7077 0.7077 0.4629 0.5141 0.7181 -0.1450 + 0.3831i -0.1450 - 0.3831i 0.3106 0.5300 -0.4077 -0.3302 - 0.3239i -0.3302 + 0.3239i -0.7914 0.5455 0.0944 -0.1645 - 0.2465i -0.1645 + 0.2465i 0.2091 d的每一列即为一个特征向量v = 3.2037 0 0 0 0 0 0.5684 0 0 0 0 0 -0.1026 + 0.5388i 0 0 0 0 0 -0.1026 - 0.5388i 0 0 0 0 0 -0.0457 V中非零值即为每一列特征向量对应的特征值随机生成5维列向量x第一组 X=rand(1,5)X = 0.6020 0.2630 0.6541 0.6892 0.7482求X*X X*Xa