eig versus svd functions?

Question

Traian Preda 2014-7-18

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https://ww2.mathworks.cn/matlabcentral/answers/142331-eig-versus-svd-functions

编辑： Alfonso Nieto-Castanon 2014-7-18

Hi,

I would like to ask what is the difference between the function eig and svd and what is the difference between the right eigenvectors and the right singular eigenvectors of these functions?

Thank you

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Answer 1

Alfonso Nieto-Castanon 2014-7-18

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编辑：Alfonso Nieto-Castanon 2014-7-18

SVD is a decomposition for arbitrary-size matrices, while EIG applies only to square matrices. They are very much related:

The right singular vectors of A are the eigenvectors of A'*A, and the left singular vectors of A are the eigenvectors of A*A'.

Similarly the singular values of A are the square root of the eigenvalues of A*A' (or A'*A, the eigenvalues of those are just the same)

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Traian Preda 2014-7-18

Hi,

Thank you very much for the answer. So by using eig to a non-square matrix the eigenvectors I get it are wrong? Should I use the svd function to get the correct eigenvectors?

Alfonso Nieto-Castanon 2014-7-18

not exactly, there are simply no "eigenvectors" of a non-square matrix (eigenvalues/eigenvectors are only defined for square matrices)

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Answer 2

Traian Preda 2014-7-18

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Hi again,

For example for this A matrix (square) I get using eig different right eigenvectors than by using svd. Seems that the sign of the ones produces by svd are the correct ones. Why this is happen?

-0,309435400000000 0,0211961600000000 0,0136410800000000 0,00743749000000000 0,00982272000000000 0,0111470200000000 0,00231817000000000 0,00864246000000000 0,00882075000000000 0,00284796000000000 0,00218905000000000 0,000430500000000000 0,000376470000000000 -0,113049800000000 0,00313088000000000 0,00149085000000000 0,00262337000000000 0,00311209000000000 -0,000262570000000000 0,00274903000000000 0,00264655000000000 0,000447630000000000 -0,000306180000000000 -0,00343315000000000 0,0103301100000000 0,0146587200000000 -0,324318800000000 0,00848285000000000 0,0167109100000000 0,0201004200000000 -0,00347676000000000 0,00632566000000000 0,00637912000000000 0,00186275000000000 0,00111713000000000 -0,00141243000000000 0,0464070100000000 0,0700929900000000 0,0900204400000000 -1,25637600000000 0,0836402500000000 0,103855100000000 -0,0114356300000000 0,0277253900000000 0,0280522900000000 0,00843094000000000 0,00547936000000000 -0,00411460000000000 0,0343002800000000 0,0606666900000000 0,0772628000000000 0,0388068100000000 -0,983633200000000 0,101284700000000 0,000288840000000000 0,0190463800000000 0,0194692000000000 0,00636249000000000 0,00501263000000000 0,00161963000000000 0,0216479800000000 0,0390856900000000 0,0497280000000000 0,0259531400000000 0,0538816600000000 -0,620819600000000 0,000968680000000000 0,0118906800000000 0,0121738400000000 0,00402735000000000 0,00325024000000000 0,00144150000000000 0,0560331100000000 0,0322143500000000 0,0452270700000000 0,00474108000000000 0,0665512800000000 0,0879651100000000 -1,14406400000000 0,0420314600000000 0,0413540800000000 0,00940455000000000 0,000919950000000000 -0,0325435900000000 0,0651720900000000 0,148360900000000 0,0550088600000000 0,0311400300000000 0,0376528500000000 0,0420123000000000 0,0135666400000000 -1,51919300000000 0,0778850700000000 -0,0322261000000000 0,0138896200000000 0,0193712200000000 0,0669056200000000 0,140490500000000 0,0573799400000000 0,0320810700000000 0,0399603600000000 0,0448506100000000 0,0126767400000000 0,0401568300000000 -1,54203000000000 -0,119447500000000 -0,000335680000000000 0,00939138000000000 0,0818470500000000 0,154808700000000 0,0715045300000000 0,0394081700000000 0,0507693800000000 0,0573505200000000 0,0137023800000000 -0,0385798100000000 -0,184705900000000 -1,07855600000000 -0,0214767400000000 -0,00365997000000000 0,0835618100000000 0,118544600000000 0,0760379600000000 0,0406107300000000 0,0561993500000000 0,0643053400000000 0,00980772000000000 0,0875483300000000 0,0746029100000000 -0,0136161900000000 -1,29450700000000 -0,0388254600000000 0,00108525000000000 0,000792020000000000 0,00104497000000000 0,000534560000000000 0,000812510000000000 0,000944030000000000 4,82500000000000e-05 0,000902380000000000 0,000866900000000000 0,000141640000000000 -0,000112090000000000 -0,174199600000000

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Alfonso Nieto-Castanon 2014-7-18

the eigenvectors of a square matrix are not generally the same as any of the singular vectors of that same matrix (they are equal/equivalent only when the matrix is symmetric)

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Answer 3

Traian Preda 2014-7-18

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OK, then this can be reason that when I try to rebuilt the A matrix, I cannot succeed by using the V, D eigenvectors ([V,D]=eig(A)), but I can rebuilt it when I use V,S,D ([V,S,D]=svd(A))?

Thank you very much