使用 Excel 和 Python 时矩阵的逆矩阵不同
Inverse of matrix different when using Excel and Python
我有这个矩阵
array([[ 4.58437363e-04, 6.87656045e-03, 6.87656045e-02,
1.10652674e-01, 1.65979011e+00, 1.65979011e+01,
3.66749891e-03, 5.50124836e-02, 5.50124836e-01,
8.85221390e-01, 1.32783208e+01, 1.32783208e+02],
[ 6.99837609e-02, 1.39967522e+00, 1.43466710e+01,
1.46232455e-01, 2.92464911e+00, 2.99776533e+01,
2.53691133e-01, 5.07382267e+00, 5.20066823e+01,
5.30092650e-01, 1.06018530e+01, 1.08668993e+02],
[ 2.84535496e-03, 3.41442595e-02, 4.83710343e-01,
1.84948072e-01, 2.21937687e+00, 3.14411723e+01,
1.23061602e-02, 1.47673922e-01, 2.09204723e+00,
7.99900413e-01, 9.59880495e+00, 1.35983070e+02],
[ 3.31593931e-02, 2.32115752e-01, 6.46608166e+00,
2.42778797e-01, 1.69945158e+00, 4.73418654e+01,
8.70102475e-02, 6.09071733e-01, 1.69669983e+01,
6.37051563e-01, 4.45936094e+00, 1.24225055e+02]])
当我使用数组公式
计算Excel中的逆时
=MINVERSE(B27:M38)
我明白了
-1.68568E+17 -1.32003E+15 -8.52171E+14 4.94857E+16 -7.86173E+15 4.96639E+14 -1.13617E+17 2.33346E+15 6.79271E+14 -4.96618E+16 2.85561E+15 4.09202E+13
-3.06571E+16 -5.59088E+14 2.25152E+14 1.31473E+15 3.07834E+14 -5.62038E+13 4.52019E+15 1.70789E+14 -2.28526E+13 4.20304E+15 -2.45416E+14 -1.47953E+12
1.81455E+14 5.59212E+13 -6.6655E+12 2.67828E+14 -3.69694E+13 -1.67831E+12 1.97479E+14 2.98161E+13 -2.22862E+12 5.85569E+13 -7.57931E+12 8.07625E+11
7.85428E+13 -5.42549E+14 2.63919E+13 -2.54175E+16 7.29192E+14 9.38632E+13 1.42654E+16 -3.63692E+13 -6.24915E+13 5.70179E+15 -1.68685E+14 -2.09159E+13
-6.01158E+14 5.7767E+13 -3.66461E+13 1.08182E+15 -1.46389E+14 4.3594E+12 -6.35683E+14 1.32624E+13 9.66142E+12 -9.67009E+14 4.0771E+13 2.74315E+12
-1.03323E+14 -1.55034E+13 2.05109E+12 5.20197E+13 6.17586E+12 -7.19368E+11 6.94553E+13 6.31804E+11 -2.80104E+11 4.07483E+13 -1.84912E+12 -1.18293E+11
5.18468E+16 4.88032E+14 -2.99218E+14 -8.71756E+15 1.28383E+15 1.89111E+14 6.82672E+15 -3.5848E+15 1.508E+14 -2.18052E+16 1.26861E+15 -1.32931E+13
7.49045E+15 2.11314E+14 -2.67962E+13 -3.9188E+14 -5.12058E+13 7.56573E+12 -1.57588E+15 3.06765E+13 -5.03822E+12 -2.60028E+14 1.91038E+13 -1.27174E+11
1.51294E+14 -1.5578E+13 1.14145E+12 -1.22333E+14 1.23247E+13 -3.1036E+11 6.15507E+13 -1.9115E+12 -1.0148E+11 7.22411E+13 -3.48559E+12 -1.47163E+11
2.16878E+16 6.9913E+14 -7.80275E+13 -1.01657E+15 -2.05019E+14 1.06245E+13 -5.32103E+15 6.6262E+13 -1.03726E+13 -1.95913E+15 4.15017E+13 1.10843E+13
-9.77565E+14 -7.57216E+13 4.71499E+12 -1.08066E+14 2.16929E+13 -4.34593E+11 5.51293E+14 -4.82981E+11 -4.12064E+11 1.55974E+14 -4.4425E+12 -7.30823E+11
-2.95506E+13 4.54732E+12 2.37753E+11 1.9272E+13 -3.87232E+11 -75168166104 -3.20143E+13 -5.46016E+11 75013747465 40232740147 7907929619 -2339538070
但是，当我使用 python 使用 numpy
计算逆时
ata_inv=numpy.linalg.inv(ata)
我明白了
array([[ -2.92174491e+17, 8.78092304e+13, -3.74278555e+14,
6.85675722e+16, -7.88519831e+15, 3.97491106e+14,
-1.19443836e+17, 1.96271342e+15, 7.75339916e+14,
-4.31451830e+16, 2.66377305e+15, 1.34678664e+13],
[ -1.20148124e+16, -1.39577173e+14, 1.62609200e+14,
-2.05343938e+15, 2.47311333e+14, -3.70464968e+13,
2.13473015e+15, 2.02356660e+14, -3.77977161e+13,
3.19465019e+15, -2.20164429e+14, 3.94769138e+12],
[ 1.03452264e+14, 4.49963139e+13, -5.83232093e+12,
3.62478360e+14, -3.57104899e+13, -2.31688528e+12,
2.19120097e+14, 3.04727893e+13, -2.15230839e+12,
6.27499323e+13, -8.08233158e+12, 8.14399730e+11],
[ 8.99729769e+15, -7.33200660e+14, -2.07714396e+13,
-2.58353689e+16, 6.90934358e+14, 9.91927528e+13,
1.47797299e+16, -3.93583461e+12, -6.50452782e+13,
4.86796183e+15, -1.39660309e+14, -1.81127868e+13],
[ -3.48430812e+15, 1.12742047e+12, -2.70225162e+13,
1.67561520e+15, -1.40107200e+14, 1.14260887e+12,
-3.44439537e+14, 8.36705528e+12, 1.21738153e+13,
-8.28685468e+14, 3.72852679e+13, 1.98978705e+12],
[ 6.58910302e+13, -1.22902337e+13, 1.39683509e+12,
2.27296720e+13, 5.47359894e+12, -5.34359515e+11,
5.07774800e+13, 9.33372125e+11, -3.97227009e+11,
3.00641665e+13, -1.53501520e+12, -6.79505533e+10],
[ 4.59914066e+15, -9.70909800e+14, -2.45271675e+14,
-3.96303832e+15, 1.32677231e+15, 1.79898008e+14,
1.71089473e+16, -3.68415009e+15, 2.01159781e+14,
-2.02750198e+16, 1.29185320e+15, -2.94060796e+13],
[ 5.29168020e+15, 1.89585534e+14, -1.64697929e+13,
-9.16154758e+12, -4.07498605e+13, 4.84728269e+12,
-1.45007322e+15, 2.63347736e+13, -3.81669434e+12,
-9.78100475e+13, 1.34239556e+13, -7.53836147e+11],
[ 2.82268043e+14, -1.59196485e+13, 7.63324745e+11,
-1.48459210e+14, 1.26222499e+13, -2.02131815e+11,
6.36919902e+13, -1.56430001e+12, -2.38728387e+11,
6.81867356e+13, -3.41941515e+12, -1.22099795e+11],
[ 1.52015592e+16, 6.32482966e+14, -4.52320855e+13,
-5.91644074e+12, -1.65167701e+14, 2.55304641e+12,
-4.89017768e+15, 5.33549362e+13, -7.52567672e+12,
-1.41538267e+15, 2.19297078e+13, 9.07563720e+12],
[ -4.64671981e+14, -7.43091440e+13, 2.03611340e+12,
-1.63922466e+14, 1.83754071e+13, 9.08311874e+10,
5.27684329e+14, 7.77828696e+11, -5.83508093e+11,
1.08824465e+14, -2.76885598e+12, -5.61136635e+11],
[ -2.89682527e+13, 5.34539735e+12, 2.84749412e+11,
1.45849833e+13, -2.84208676e+11, -6.06452748e+10,
-3.44726894e+13, -5.87172107e+11, 5.76165423e+10,
1.36639289e+12, -4.78031299e+10, -4.71743402e+09]])
为什么这些答案不同？哪个是正确的？
最后，我只想求一个多项式的最小二乘估计。我以为我会用逆来做，但我现在认为这不是最好的方法。
您得到不同答案的原因有两个
Fist，存在数值不稳定的问题。看看倒数中的值——它们非常大。这意味着你要求逆的原始矩阵至少有一个非常小的奇异值，这意味着矩阵很难求逆。
其次，您正在尝试反转非方阵。矩阵逆实际上仅在方阵的情况下是唯一的。对于非方阵，正式的矩阵逆不存在。相反，您调用的函数将找到左逆或右逆，但这些通常不是唯一的，因此找到这些逆的两种不同但完全有效的算法实现可能会给出不同的答案。
我有这个矩阵
array([[ 4.58437363e-04, 6.87656045e-03, 6.87656045e-02,
1.10652674e-01, 1.65979011e+00, 1.65979011e+01,
3.66749891e-03, 5.50124836e-02, 5.50124836e-01,
8.85221390e-01, 1.32783208e+01, 1.32783208e+02],
[ 6.99837609e-02, 1.39967522e+00, 1.43466710e+01,
1.46232455e-01, 2.92464911e+00, 2.99776533e+01,
2.53691133e-01, 5.07382267e+00, 5.20066823e+01,
5.30092650e-01, 1.06018530e+01, 1.08668993e+02],
[ 2.84535496e-03, 3.41442595e-02, 4.83710343e-01,
1.84948072e-01, 2.21937687e+00, 3.14411723e+01,
1.23061602e-02, 1.47673922e-01, 2.09204723e+00,
7.99900413e-01, 9.59880495e+00, 1.35983070e+02],
[ 3.31593931e-02, 2.32115752e-01, 6.46608166e+00,
2.42778797e-01, 1.69945158e+00, 4.73418654e+01,
8.70102475e-02, 6.09071733e-01, 1.69669983e+01,
6.37051563e-01, 4.45936094e+00, 1.24225055e+02]])
当我使用数组公式
计算Excel中的逆时=MINVERSE(B27:M38)
我明白了
-1.68568E+17 -1.32003E+15 -8.52171E+14 4.94857E+16 -7.86173E+15 4.96639E+14 -1.13617E+17 2.33346E+15 6.79271E+14 -4.96618E+16 2.85561E+15 4.09202E+13
-3.06571E+16 -5.59088E+14 2.25152E+14 1.31473E+15 3.07834E+14 -5.62038E+13 4.52019E+15 1.70789E+14 -2.28526E+13 4.20304E+15 -2.45416E+14 -1.47953E+12
1.81455E+14 5.59212E+13 -6.6655E+12 2.67828E+14 -3.69694E+13 -1.67831E+12 1.97479E+14 2.98161E+13 -2.22862E+12 5.85569E+13 -7.57931E+12 8.07625E+11
7.85428E+13 -5.42549E+14 2.63919E+13 -2.54175E+16 7.29192E+14 9.38632E+13 1.42654E+16 -3.63692E+13 -6.24915E+13 5.70179E+15 -1.68685E+14 -2.09159E+13
-6.01158E+14 5.7767E+13 -3.66461E+13 1.08182E+15 -1.46389E+14 4.3594E+12 -6.35683E+14 1.32624E+13 9.66142E+12 -9.67009E+14 4.0771E+13 2.74315E+12
-1.03323E+14 -1.55034E+13 2.05109E+12 5.20197E+13 6.17586E+12 -7.19368E+11 6.94553E+13 6.31804E+11 -2.80104E+11 4.07483E+13 -1.84912E+12 -1.18293E+11
5.18468E+16 4.88032E+14 -2.99218E+14 -8.71756E+15 1.28383E+15 1.89111E+14 6.82672E+15 -3.5848E+15 1.508E+14 -2.18052E+16 1.26861E+15 -1.32931E+13
7.49045E+15 2.11314E+14 -2.67962E+13 -3.9188E+14 -5.12058E+13 7.56573E+12 -1.57588E+15 3.06765E+13 -5.03822E+12 -2.60028E+14 1.91038E+13 -1.27174E+11
1.51294E+14 -1.5578E+13 1.14145E+12 -1.22333E+14 1.23247E+13 -3.1036E+11 6.15507E+13 -1.9115E+12 -1.0148E+11 7.22411E+13 -3.48559E+12 -1.47163E+11
2.16878E+16 6.9913E+14 -7.80275E+13 -1.01657E+15 -2.05019E+14 1.06245E+13 -5.32103E+15 6.6262E+13 -1.03726E+13 -1.95913E+15 4.15017E+13 1.10843E+13
-9.77565E+14 -7.57216E+13 4.71499E+12 -1.08066E+14 2.16929E+13 -4.34593E+11 5.51293E+14 -4.82981E+11 -4.12064E+11 1.55974E+14 -4.4425E+12 -7.30823E+11
-2.95506E+13 4.54732E+12 2.37753E+11 1.9272E+13 -3.87232E+11 -75168166104 -3.20143E+13 -5.46016E+11 75013747465 40232740147 7907929619 -2339538070
但是，当我使用 python 使用 numpy
ata_inv=numpy.linalg.inv(ata)
我明白了
array([[ -2.92174491e+17, 8.78092304e+13, -3.74278555e+14,
6.85675722e+16, -7.88519831e+15, 3.97491106e+14,
-1.19443836e+17, 1.96271342e+15, 7.75339916e+14,
-4.31451830e+16, 2.66377305e+15, 1.34678664e+13],
[ -1.20148124e+16, -1.39577173e+14, 1.62609200e+14,
-2.05343938e+15, 2.47311333e+14, -3.70464968e+13,
2.13473015e+15, 2.02356660e+14, -3.77977161e+13,
3.19465019e+15, -2.20164429e+14, 3.94769138e+12],
[ 1.03452264e+14, 4.49963139e+13, -5.83232093e+12,
3.62478360e+14, -3.57104899e+13, -2.31688528e+12,
2.19120097e+14, 3.04727893e+13, -2.15230839e+12,
6.27499323e+13, -8.08233158e+12, 8.14399730e+11],
[ 8.99729769e+15, -7.33200660e+14, -2.07714396e+13,
-2.58353689e+16, 6.90934358e+14, 9.91927528e+13,
1.47797299e+16, -3.93583461e+12, -6.50452782e+13,
4.86796183e+15, -1.39660309e+14, -1.81127868e+13],
[ -3.48430812e+15, 1.12742047e+12, -2.70225162e+13,
1.67561520e+15, -1.40107200e+14, 1.14260887e+12,
-3.44439537e+14, 8.36705528e+12, 1.21738153e+13,
-8.28685468e+14, 3.72852679e+13, 1.98978705e+12],
[ 6.58910302e+13, -1.22902337e+13, 1.39683509e+12,
2.27296720e+13, 5.47359894e+12, -5.34359515e+11,
5.07774800e+13, 9.33372125e+11, -3.97227009e+11,
3.00641665e+13, -1.53501520e+12, -6.79505533e+10],
[ 4.59914066e+15, -9.70909800e+14, -2.45271675e+14,
-3.96303832e+15, 1.32677231e+15, 1.79898008e+14,
1.71089473e+16, -3.68415009e+15, 2.01159781e+14,
-2.02750198e+16, 1.29185320e+15, -2.94060796e+13],
[ 5.29168020e+15, 1.89585534e+14, -1.64697929e+13,
-9.16154758e+12, -4.07498605e+13, 4.84728269e+12,
-1.45007322e+15, 2.63347736e+13, -3.81669434e+12,
-9.78100475e+13, 1.34239556e+13, -7.53836147e+11],
[ 2.82268043e+14, -1.59196485e+13, 7.63324745e+11,
-1.48459210e+14, 1.26222499e+13, -2.02131815e+11,
6.36919902e+13, -1.56430001e+12, -2.38728387e+11,
6.81867356e+13, -3.41941515e+12, -1.22099795e+11],
[ 1.52015592e+16, 6.32482966e+14, -4.52320855e+13,
-5.91644074e+12, -1.65167701e+14, 2.55304641e+12,
-4.89017768e+15, 5.33549362e+13, -7.52567672e+12,
-1.41538267e+15, 2.19297078e+13, 9.07563720e+12],
[ -4.64671981e+14, -7.43091440e+13, 2.03611340e+12,
-1.63922466e+14, 1.83754071e+13, 9.08311874e+10,
5.27684329e+14, 7.77828696e+11, -5.83508093e+11,
1.08824465e+14, -2.76885598e+12, -5.61136635e+11],
[ -2.89682527e+13, 5.34539735e+12, 2.84749412e+11,
1.45849833e+13, -2.84208676e+11, -6.06452748e+10,
-3.44726894e+13, -5.87172107e+11, 5.76165423e+10,
1.36639289e+12, -4.78031299e+10, -4.71743402e+09]])
为什么这些答案不同？哪个是正确的？
最后，我只想求一个多项式的最小二乘估计。我以为我会用逆来做，但我现在认为这不是最好的方法。
您得到不同答案的原因有两个
Fist，存在数值不稳定的问题。看看倒数中的值——它们非常大。这意味着你要求逆的原始矩阵至少有一个非常小的奇异值，这意味着矩阵很难求逆。
其次，您正在尝试反转非方阵。矩阵逆实际上仅在方阵的情况下是唯一的。对于非方阵，正式的矩阵逆不存在。相反，您调用的函数将找到左逆或右逆，但这些通常不是唯一的，因此找到这些逆的两种不同但完全有效的算法实现可能会给出不同的答案。