大连理工大学线性代数实验上机报告

1、大连理工大学线性代数实验上机报告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 =0.81470.09750.15760.14190.65570.90580.27850.97060.42180.03570.12700.54690.95720.91570.84910.91340.95750.48540.79220.93400.63240.96490.80030.95950.6787>> B=rand(5) 随机生成五阶方阵 B B =0.75770.70600.82350.43870.48980.74310.03180.69480.38160.44

2、560.39220.27690.31710.76550.64630.65550.04620.95020.79520.70940.1712 0.0971 0.0344 0.1869 0.7547>> b=rand(1,5)' 随机生成列向量 b b =0.27600.67970.65510.16260.1190计算 A+B>> A+Bans =1.57250.80360.98110.58061.14551.64890.31031.66540.80330.48130.51920.82381.27431.68131.49541.56891.00371.43561.58

3、741.64340.80351.06200.83471.14641.4334计算 A-B >> A-BqV=x «搦阴q=xv來9L8SCOZLLC68SSCZ8ZZL09£L£0ZL6西眈P9Z9£P176Z6S61706C17£Z9£170LSC01/乙1789SCZ£Z0£01700CZ96LS17606217£OL£9ZSZS6£l7L£890CS88S0C=sue09Z00-9SZZ0899Z08Z980即 91/0917220ocooo-8mo-

4、63;14606Z9S082020209 L0L0W000Z2029920660170-SOWO89ZS0Z917S0ZS9L0099L069620-89990-98090-0Z900=sue-0.98502.43963.3124-5.65151.7085 验证克莱姆法则 >> c=A(:,1)c =0.81470.90580.12700.91340.6324>> d=A(:,2)d =0.09750.27850.54690.95750.9649 >> e=A(:,3) e =0.15760.97060.95720.48540.8003 >> f

5、=A(:,4) f =0.14190.4218 0.91570.7922 0.9595>> g=A(:,5)g =0.65570.03570.84910.93400.6787>> B1=b'd'e'f'g''B1 =0.27600.09750.15760.14190.65570.67970.27850.97060.42180.03570.65510.54690.95720.91570.84910.16260.95750.48540.79220.93400.11900.96490.80030.95950.6787>&g

6、t; B2=c'b'e'f'g''B2 =0.81470.27600.15760.14190.65570.90580.67970.97060.42180.03570.12700.65510.95720.91570.84910.91340.16260.48540.79220.93400.63240.11900.80030.95950.6787>> B3=c'd'b'f'g''B3 =0.81470.09750.27600.14190.65570.90580.27850.67970.4218

7、0.03570.12700.54690.65510.91570.84910.91340.95750.16260.79220.93400.63240.96490.11900.95950.6787>> B4=c'd'e'b'g''B4 =0.81470.09750.15760.27600.65570.90580.27850.97060.67970.03570.12700.54690.95720.65510.84910.91340.95750.48540.16260.93400.63240.96490.80030.11900.6787>

8、;> B5=c'd'e'f'b''B5 =0.81470.09750.15760.14190.27600.90580.27850.97060.42180.67970.12700.54690.95720.91570.65510.91340.95750.48540.79220.16260.63240.96490.80030.95950.1190>> x1=det(B1)/det(A) x1 =-0.9850>> x2=det(B2)/det(A) x2 =2.4396>> x3=det(B3)/det(A)

9、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 5.1680-8.60763.53142.890713.7204-14.3665-6.28243.72203.613210.0084-12.419013.6173-6.88

10、22-6.3938-23.528827.5825-2.52921.07292.41935.8870-7.2671求 B 的逆>> inv(B)ans =-0.44303.49971.3255-2.6005-0.46971.4047-1.16260.2422-0.4475-0.01190.7210-1.8189-2.06352.44340.0765-0.6122-0.18372.01650.0375-1.25640.0384-0.5157-0.73700.52671.7407求 A 的秩 >> rank(A) ans =5 求 B 的秩>> rank(B)an

11、s =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 -137.34426.8291-1.2718-2.2922-8.99518.697263.9620-31.4202-29.5061-105.6918 122.3246-9.31965.74524.625911.9660-15.402811.9582-6.3521-3.3817-16.757418.6360>> rank(A*B)ans =5>> det(A)*

12、det(B)ans =-0.0016验证 AB T BTAT, AB 1 B 1A 1,AB BA(1)>> (A*B)'ans =0.95691.55661.62372.27322.25520.69220.94010.49690.93710.80900.94611.64921.68752.35632.38000.75071.58871.88401.94212.14811.13991.52122.21492.45452.4497>> B'*A'ans =0.95691.55661.62372.27322.25520.69220.94010.496

13、90.93710.80900.94611.64921.68752.35632.38000.75071.58871.88401.94212.14811.13991.52122.21492.45452.4497(2)>> inv(B)*inv(A)ans =74.0649 35.0433 31.2288 121.5740 -137.34426.8291-1.2718-2.2922-8.99518.697263.9620-31.4202-29.5061-105.6918 122.3246-9.31965.74524.625911.9660-15.402811.9582-6.3521-3.

14、3817-16.757418.63603)>> A*Bans =0.95690.69220.94610.75071.13991.55660.94011.64921.58871.52121.62370.49691.68751.88402.21492.27320.93712.35631.94212.45452.25520.80902.38002.14812.4497>> B*Aans =2.07191.61352.19781.97691.96351.35281.25661.35491.48501.73721.71861.64541.52291.68941.69001.871

15、42.04232.01132.29322.46250.87970.96970.84890.96900.8317求矩阵 X 使得 AXB=C 首先随机生成五阶方阵 C >> C=rand(5)0.4984 0.7513 0.9593 0.8407 0.35000.9597 0.2551 0.5472 0.2543 0.19660.34040.50600.13860.81430.25110.58530.69910.14930.24350.61600.22380.89090.25750.92930.4733X=A的逆 *B 的逆>> X=inv(A)*C*inv(B)3.84

16、32-13.88582.14189.4404-4.5871-9.331241.9602-7.910128.468314.8942-7.873835.1218-5.410722.886110.158116.7545-75.607914.678449.3951-24.7450-3.556817.0848-2.901811.26705.4559实验二1. 验证：对于一般的方阵A,B,C,D ，AD B C首先随机生成方阵 A,B,C,DA=rand(5)0.82580.10670.86870.43140.13610.53830.96190.08440.91060.86930.99610.00460.

17、39980.18180.57970.07820.77490.25990.26380.54990.44270.81730.80010.14550.1450>> B=rand(5)0.85300.07600.41730.48930.78030.62210.23990.04970.33770.38970.35100.12330.90270.90010.24170.51320.18390.94480.36920.40390.40180.24000.49090.11120.0965C=rand(5)0.1320 0.2348 0.1690 0.5470 0.18350.94210.35320

18、.64910.29630.36850.95610.82120.73170.74470.62560.57520.01540.64770.18900.78020.05980.04300.45090.68680.0811>> D=rand(5)0.92940.30630.64430.93900.20770.77570.50850.37860.87590.30120.48680.51080.81160.55020.47090.43590.81760.53280.62250.23050.44680.79480.35070.58700.8443Z=A,B;C,D0.8258 0.1067 0.

19、8687 0.4314 0.13610.85300.07600.41730.48930.53830.96190.08440.91060.62210.23990.04970.33770.99610.00460.39980.18180.35100.12330.90270.90010.07820.77490.25990.26380.51320.18390.94480.36920.44270.81730.80010.14550.40180.24000.49090.11120.13200.23480.16900.54700.92940.30630.64430.93900.94210.35320.6491

20、0.29630.77570.50850.37860.87590.95610.82120.73170.74470.48680.51080.81160.55020.57520.01540.64770.18900.43590.81760.53280.62250.05980.04300.45090.68680.44680.79480.35070.58700.7803求 Z 的行列式0.86930.38970.57970.24170.54990.40390.14500.09650.18350.20770.36850.30120.62560.47090.78020.23050.08110.8443>

21、> 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 0000 0.2259 0000 0 0.170700000 0.2277 0000 0 0.4357随机生成对角矩阵 B >> B=diag(rand rand rand rand rand)0.3111 00000 0.92340000 0 0.4

22、30200000 0.1848 0000 0 0.9049随机生成对角矩阵C>> C=diag(rand rand rand rand rand)C =0.9797 00000 0.43890000 0 0.111100000 0.2581 0000 0 0.4087随机生成对角矩阵D>> D=diag(rand rand rand rand rand)0 0.2622 0 0 0000.6028 0 0000 0.7112 0000 0 0.2217>> Z=A,B;C,DZ =0.194800000.3111000000.225900000.923400

23、0000.170700000.4302000000.227700000.1848000000.435700000.90490.9797 0 0 0 00 0.4389 0 0 00 0.2622 0 0 00.11110 00 00 00 00 00 0 计算 Z 的行列式 >> det(Z)ans =-1.1243e-0040 00.6028 0 00 0.2581 00 0.7112 00 0 0.40870 0 0.2217计算 det(A)*det(D)-det(B)*det(C)>> det(A)*det(D)-det(B)*det(C)ans =-9.310

24、7e-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)1.0000 0 0 0 -0.35480.46560 1.0000 0 0 -1.4905-2.00200 0 1.00

25、00 0 0.04730.39500 0 0 1.0000 1.79841.3383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=

26、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 =1.20430.03040.35171.2940>> x2=A2/b1x2 =1.20430.01820.35171.2940>> x3=A3/b1x3 =1.2043

27、0.00610.35171.2940>> x4=A4/b1x4 =1.20430.00300.35171.2940>> x5=A2/b1x5 =1.20430.01820.35171.2940>> x6=A2/b2x6 =1.20310.01820.35141.2931>> x5=A1/b2x5 =1.20310.03040.35141.2931>> x6=A2/b2x6 =1.20310.01820.35141.2931>> x7=A3/b2x7 =1.20310.00610.35141.2931>> x8=

28、A4/b2 x8 =1.20310.00300.35141.2931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 =0.11740.50790.02920.57850.54680.29670.08550.92890.23730.52110.31880.26250.73030.45880.23160.42420.80100.48860.96310.4889正交化>> b=null(a)0.6777-0.5941-0.30090.28920.1163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

29、)ans =3.20370.5684-0.1026 + 0.5388i-0.1026 - 0.5388i-0.0457>> d,v=eig(A)0.2283-0.53540.0179 + 0.1710i0.0179- 0.1710i-0.13830.32430.14990.70770.70770.46290.51410.7181-0.1450 + 0.3831i-0.1450- 0.3831i0.31060.5300-0.4077-0.33020.3239i-0.3302 + 0.3239i-0.79140.54550.0944-0.1645 -0.2465i-0.1645 + 0.2465i0.2091d 的每一列即为一个特征向量3.2037000000.568400000-0.1026 + 0.5388i00000-0.10260.5388i00000-0.0457V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)'0.60200.26300.65410.68920.7482 求 X*X'>> X*X'ans =0.36240.15830.39370.41490.45040.15830.06920.17200.18120.19670.