我们可以从矩阵中获得特征向量的不同解决方案吗? [英] Could we get different solutions for eigenVectors from a matrix?
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问题描述
我的目的是找到矩阵的特征向量.在Matlab中，有一个[V,D] = eig(M)通过使用:[V,D] = eig(M)来获取矩阵的特征向量.另外，我使用网站 WolframAlpha 仔细检查了结果.
My purpose is to find a eigenvectors of a matrix. In Matlab, there is a [V,D] = eig(M) to get the eigenvectors of matrix by using: [V,D] = eig(M). Alternatively I used the website WolframAlpha to double check my results.
我们有一个名为M的10X10矩阵:
We have a 10X10 matrix called M:
0.736538062307847 -0.638137874226607 -0.409041107160722 -0.221115060391256 -0.947102932298308 0.0307937582853794 1.23891356582639 1.23213871779652 0.763885436104244 -0.805948245321096
-1.00495215920171 -0.563583317483057 -0.250162608745252 0.0837145788064272 -0.201241986127792 -0.0351472158148094 -1.36303599752928 0.00983020375259212 -0.627205458137858 0.415060573134481
0.372470672825535 -0.356014310976260 -0.331871925811400 0.151334279460039 0.0983275066581362 -0.0189726910991071 0.0261595600177302 -0.752014960080128 -0.00643718050231003 0.802097123260581
1.26898635468390 -0.444779390923673 0.524988731629985 0.908008064819586 -1.66569084499144 -0.197045800083481 1.04250295411159 -0.826891197039745 2.22636770820512 0.226979917020922
-0.307384714237346 0.00930402052877782 0.213893752473805 -1.05326116146192 -0.487883985126739 0.0237598951768898 -0.224080566774865 0.153775526014521 -1.93899137944122 -0.300158630162419
7.04441299430365 -1.34338456640793 -0.461083493351887 5.30708311554706 -3.82919170270243 -2.18976040860706 6.38272280044908 2.33331906669527 9.21369926457948 -2.11599193328696
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0
D:
2.84950796497613 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 1.08333535157800 + 0.971374792725758i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 1.08333535157800 - 0.971374792725758i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -2.05253164206377 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -0.931513274011512 + 0.883950434279189i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -0.931513274011512 - 0.883950434279189i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -1.41036956613286 + 0.354930202789307i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -1.41036956613286 - 0.354930202789307i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i -0.374014257422547 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i
0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.00000000000000 + 0.00000000000000i 0.165579401742139 + 0.00000000000000i
V:
-0.118788118233448 + 0.00000000000000i 0.458452024790792 + 0.00000000000000i 0.458452024790792 + -0.00000000000000i -0.00893883603500744 + 0.00000000000000i -0.343151745490688 - 0.0619235203325516i -0.343151745490688 + 0.0619235203325516i -0.415371644459693 + 0.00000000000000i -0.415371644459693 + -0.00000000000000i -0.0432672840354827 + 0.00000000000000i 0.0205670999343567 + 0.00000000000000i
0.0644460666316380 + 0.00000000000000i -0.257319460426423 + 0.297135138351391i -0.257319460426423 - 0.297135138351391i 0.000668740843331284 + 0.00000000000000i -0.240349418297316 + 0.162117384568559i -0.240349418297316 - 0.162117384568559i -0.101240986260631 + 0.370051721507625i -0.101240986260631 - 0.370051721507625i 0.182133003667802 + 0.00000000000000i 0.0870047828436781 + 0.00000000000000i
-0.0349638967773464 + 0.00000000000000i -0.0481533171088709 - 0.333551383088345i -0.0481533171088709 + 0.333551383088345i -5.00304864960391e-05 + 0.00000000000000i -0.0491721720673945 + 0.235973015480054i -0.0491721720673945 - 0.235973015480054i 0.305000451960374 + 0.180389787086258i 0.305000451960374 - 0.180389787086258i -0.766686233364027 + 0.00000000000000i 0.368055402163444 + 0.00000000000000i
-0.328483258287378 + 0.00000000000000i -0.321235466934363 - 0.0865401147007471i -0.321235466934363 + 0.0865401147007471i -0.0942807049530764 + 0.00000000000000i -0.0354015249204485 + 0.395526630779543i -0.0354015249204485 - 0.395526630779543i -0.0584777280581259 - 0.342389123727367i -0.0584777280581259 + 0.342389123727367i 0.0341847135233905 + 0.00000000000000i -0.00637190625187862 + 0.00000000000000i
0.178211880664383 + 0.00000000000000i 0.236391683569043 - 0.159628238798322i 0.236391683569043 + 0.159628238798322i 0.00705341924756006 + 0.00000000000000i 0.208292766328178 + 0.256171148954103i 0.208292766328178 - 0.256171148954103i -0.319285221542254 - 0.0313551221105837i -0.319285221542254 + 0.0313551221105837i -0.143900055026164 + 0.00000000000000i -0.0269550068563120 + 0.00000000000000i
-0.908350536903352 + 0.00000000000000i 0.208752559894992 + 0.121276611951418i 0.208752559894992 - 0.121276611951418i -0.994408141243082 + 0.00000000000000i 0.452243212306010 + 0.00000000000000i 0.452243212306010 + -0.00000000000000i 0.273997199582534 - 0.0964058973906923i 0.273997199582534 + 0.0964058973906923i -0.0270087356931836 + 0.00000000000000i 0.00197408431000798 + 0.00000000000000i
-0.0416872385315279 + 0.00000000000000i 0.234583850413183 - 0.210340074973091i 0.234583850413183 + 0.210340074973091i 0.00435502958971167 + 0.00000000000000i 0.160642433241717 + 0.218916331789935i 0.160642433241717 - 0.218916331789935i 0.276971588308683 + 0.0697020017773242i 0.276971588308683 - 0.0697020017773242i 0.115683515205146 + 0.00000000000000i 0.124212913671392 + 0.00000000000000i
0.0226165595687948 + 0.00000000000000i 0.00466011130798999 + 0.270099580217056i 0.00466011130798999 - 0.270099580217056i -0.000325812684017280 + 0.00000000000000i 0.222664282388928 + 0.0372585184944646i 0.222664282388928 - 0.0372585184944646i 0.129604953142137 - 0.229763189016417i 0.129604953142137 + 0.229763189016417i -0.486968076893485 + 0.00000000000000i 0.525456559984271 + 0.00000000000000i
-0.115277185508808 + 0.00000000000000i -0.204076984892299 + 0.103102999488027i -0.204076984892299 - 0.103102999488027i 0.0459338618810664 + 0.00000000000000i 0.232009172507840 - 0.204443701767505i 0.232009172507840 + 0.204443701767505i -0.0184618718969471 + 0.238119465887194i -0.0184618718969471 - 0.238119465887194i -0.0913994930540061 + 0.00000000000000i -0.0384824814248494 + 0.00000000000000i
-0.0146296269545178 + 0.00000000000000i 0.0235283849818557 - 0.215256480570249i 0.0235283849818557 + 0.215256480570249i -0.00212178438590738 + 0.00000000000000i 0.0266030060993678 - 0.209766836873709i 0.0266030060993678 + 0.209766836873709i -0.172989400304240 - 0.0929551855455724i -0.172989400304240 + 0.0929551855455724i -0.309302420721495 + 0.00000000000000i 0.750171291624984 + 0.00000000000000i
我得到了以下结果:
- 原始矩阵:
- WolframAlpha的结果:
- Matlab Eig的结果:
D(特征值)
V(特征向量)
是否有可能为eigenVectors获得不同的解决方案，或者它应该是唯一的答案.我有兴趣澄清这个概念.
Is it possible to get different solutions for eigenVectors or it should be a unique answer. I am interested to get clarified on this concept.
推荐答案
本征向量不是唯一的，原因有很多.更改符号，对于相同的特征值，特征向量仍然是特征向量.实际上，乘以任何常数，特征向量仍然是该常数.有时不同的工具可以选择不同的规范化.
Eigenvectors are NOT unique, for a variety of reasons. Change the sign, and an eigenvector is still an eigenvector for the same eigenvalue. In fact, multiply by any constant, and an eigenvector is still that. Different tools can sometimes choose different normalizations.
如果特征值的多重性大于1，则特征向量又不是唯一的，只要它们跨越相同的子空间即可.
If an eigenvalue is of multiplicity greater than one, then the eigenvectors are again not unique, as long as they span the same subspace.
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