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.
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
I was given the following results:
D(eigenvalues)
V(eigenvectors)
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.
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.
If an eigenvalue is of multiplicity greater than one, then the eigenvectors are again not unique, as long as they span the same subspace.