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BB=0.75770.70600.82350.43870.48980.74310.03180.69480.38160.44560.39220.27690.31710.76550.64630.65550.04620.95020.79520.70940.17120.09710.03440.18690.7547>>b=rand(1,5)'随机生成列向量bb=0.27600.67970.65510.16260.1190计算A+B>>A+Bans=1.57250.80360.98110.58061.14551.64890.31031.66540.80330.48130.51920.82381.27431.68131.49541.56891.00371.43561.58741.64340.80351.06200.83471.14641.4334计算A-B>>A-B0.79220.9595>>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>>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=0.81470.09750.27600.14190.65570.90580.27850.67970.42180.03570.12700.54690.65510.91570.84910.91340.95750.16260.79220.93400.63240.96490.11900.95950.6787>>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>>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)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=3.1375-0.8078-1.8788-4.21945.1680-8.60763.53142.890713.7204-14.3665-6.28243.72203.613210.0084-12.419013.6173-6.8822-6.3938-23.528827.5825-2.52921.07292.41935.8870-7.2671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)ans=5求A*B的行列式>>det(A*B)ans=-0.0016求A*B的逆>>inv(A*B)ans=-74.064935.043331.2288121.5740-137.34426.8291-1.2718-2.2922-8.99518.697263.9620-31.4202-29.5061-105.6918122.3246-9.31965.74524.625911.9660-15.402811.9582-6.3521-3.3817-16.757418.6360>>rank(A*B)ans=5>>det(A)*det(B)ans=-0.0016验证（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=0.95691.55661.62372.27322.25520.69220.94010.49690.93710.80900.94611.64921.68752.35632.38000.75071.58871.88401.94212.14811.13991.52122.21492.45452.4497（2）>>inv(B)*inv(A)ans=-74.064935.043331.2288121.5740-137.34426.8291-1.2718-2.2922-8.99518.697263.9620-31.4202-29.5061-105.6918122.3246-9.31965.74524.625911.9660-15.402811.9582-6.3521-3.3817-16.757418.6360（3）>>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=2.07191.61352.19781.97691.96351.35281.25661.35491.48501.73721.71861.64541.52291.68941.69001.87142.04232.01132.29322.46250.87970.96970.84890.96900.8317求矩阵X使得AXB=C首先随机生成五阶方阵C>>C=rand(5)C=0.49840.75130.95930.84070.35000.95970.25510.54720.25430.19660.34040.50600.13860.81430.25110.58530.69910.14930.24350.61600.22380.89090.25750.92930.4733X=A的逆*B的逆>>X=inv(A)*C*inv(B)X=3.8432-13.88582.14189.4404-4.5871-9.331241.9602-7.9101-28.468314.8942-7.873835.1218-5.4107-22.886110.158116.7545-75.607914.678449.3951-24.7450-3.556817.0848-2.9018-11.26705.4559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=0.82580.10670.86870.43140.13610.53830.96190.08440.91060.86930.99610.00460.39980.18180.57970.07820.77490.25990.26380.54990.44270.81730.80010.14550.1450>>B=rand(5)B=0.85300.07600.41730.48930.78030.62210.23990.04970.33770.38970.35100.12330.90270.90010.24170.51320.18390.94480.36920.40390.40180.24000.49090.11120.0965>>C=rand(5)C=0.13200.23480.16900.54700.18350.94210.35320.64910.29630.36850.95610.82120.73170.74470.62560.57520.01540.64770.18900.78020.05980.04300.45090.68680.0811>>D=rand(5)D=0.92940.30630.64430.93900.20770.77570.50850.37860.87590.30120.48680.51080.81160.55020.47090.43590.81760.53280.62250.23050.44680.79480.35070.58700.8443>>Z=[A,B;C,D]Z=0.82580.10670.86870.43140.13610.8530 0.07600.41730.48930.78030.53830.96190.08440.91060.86930.62210.23990.04970.33770.38970.99610.00460.39980.18180.57970.35100.12330.90270.90010.24170.07820.77490.25990.26380.54990.51320.18390.94480.36920.40390.44270.81730.80010.14550.14500.40180.24000.49090.11120.09650.13200.23480.16900.54700.18350.92940.30630.64430.93900.20770.94210.35320.64910.29630.36850.77570.50850.37860.87590.30120.95610.82120.73170.74470.62560.48680.51080.81160.55020.47090.57520.01540.64770.18900.78020.43590.81760.53280.62250.23050.05980.04300.45090.68680.08110.44680.79480.35070.58700.8443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([randrandrandrandrand])A=0.1948000000.2259000000.1707000000.2277000000.4357随机生成对角矩阵B>>B=diag([randrandrandrandrand])B=0.3111000000.9234000000.4302000000.1848000000.9049随机生成对角矩阵C>>C=diag([randrandrandrandrand])C=0.9797000000.4389000000.1111000000.2581000000.4087随机生成对角矩阵D>>D=diag([randrandrandrandrand])D=0.5949000000.2622000000.6028000000.7112000000.2217>>Z=[A,B;C,D]Z=0.194800000.311100000 0.22590000 0.92340000 0 0.1707000 0 0.4302000 00 0.227700 00 0.1848000000.43570 000 0.90490.979700000.5949000000.438900000.2622000000.111100000.6028000000.258100000.7112000000.408700000.2217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=1.0000000-0.35480.465601.000000-1.4905-2.0020001.000000.04730.39500001.00001.79841.3383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=[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=1.20430.00610.35171.2940>>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=A4/b2x8=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)b=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)ans=3.20370.5684-0.1026+0.5388i-0.1026-0.5388i-0.0457>>[d,v]=eig(A)d=0.2283-0.53540.0179+0.1710i0.0179-0.1710i-0.13830.32430.14990.70770.70770.46290.51410.7181-0.1450+0.3831i-0.1450-0.3831i0.31060.5300-0.4077-0.3302-0.3239i-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v=3.2037000000.568400000-0.1026+0.5388i00000-0.1026-0.5388i00000-0.0457V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)'X=0.60200.26300.65410.68920.7482求X*X'>>X*X'ans=0