eig versus svd functions?

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

Answer 1

[Alfonso Nieto-Castanon]

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)

[Traian Preda]

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]

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

Answer 2

[Traian Preda]

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

[Alfonso Nieto-Castanon]

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)

Answer 3

[Traian Preda]

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

[Alfonso Nieto-Castanon]

You can reconstruct A from its eigenvectors only if A is normal (A'*A==A*A'). You can reconstruct A from its singular vectors for any matrix A.