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How I can use the eig function for nonsymmetric matrices?

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Traian Preda
Traian Preda 2014-8-7
编辑： Chris Turnes 2014-8-7
Hi,
I have a non symmetric matrix and I try to figure out which option of the eig I should use? Thank you

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Chris Turnes
Chris Turnes 2014-8-7
The eig function does not require any additional options for nonsymmetric matrices. There is an example of this in the documentation for eig:
>> A = gallery('circul',3)
A =
1 2 3
3 1 2
2 3 1
>> [V,D] = eig(A);
>> V\A*V % verify that V diagonalizes A
ans =
6.0000 + 0.0000i 0.0000 - 0.0000i -0.0000 + 0.0000i
-0.0000 - 0.0000i -1.5000 + 0.8660i -0.0000 - 0.0000i
-0.0000 + 0.0000i -0.0000 + 0.0000i -1.5000 - 0.8660i
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Traian Preda
Traian Preda 2014-8-7
Hi,
Thank you for the answer. But for this matrix is I am using eig and I try to rebuilted back using the eigenvectors and eigenvalues I get something else than the original matrix. Therefore I think that the right eigenvectors I get using eig are wrong. I tried with svd an that it seems to work. eigenvectors I get using eig are wrong. I tried with svd an that it seems to work. Thank you
a=[-0.309042200000000 -0.00112050000000000 0.0146369900000000 0.00801360000000000 0.00951086000000000 0.0119207900000000 0.00269338000000000 0.00899029000000000 0.0120456200000000 0.00333142000000000 0.00291344000000000 0.00162112000000000;-0.00207730000000000 -0.120495000000000 0.00591404000000000 0.00262672000000000 0.00434139000000000 0.00592824000000000 -0.000548390000000000 0.00488849000000000 0.00706523000000000 0.000733070000000000 -0.00120430000000000 -0.00733420000000000;0.0106811400000000 0.00310927000000000 -0.321451300000000 0.00799418000000000 0.0146429700000000 0.0204037200000000 -0.00334158000000000 0.00629642000000000 0.00855662000000000 0.00208133000000000 0.00138921000000000 -0.000783360000000000;0.0500343300000000 0.0226932300000000 0.100244500000000 -1.21203000000000 0.0792224400000000 0.114310000000000 -0.00723418000000000 0.0281160000000000 0.0379308900000000 0.00987513000000000 0.00770613000000000 0.000929510000000000;0.0301291200000000 0.0162645000000000 0.0635045000000000 0.0263079900000000 -0.887144100000000 0.0801669700000000 -0.00166388000000000 0.0164896800000000 0.0221528100000000 0.00598660000000000 0.00502368000000000 0.00203046000000000;0.0227449100000000 0.0146744200000000 0.0508353200000000 0.0247133000000000 0.0463564800000000 -0.614663700000000 0.00122573000000000 0.0120418600000000 0.0160892700000000 0.00455634000000000 0.00414577000000000 0.00288858000000000;0.0577366400000000 -0.0267536800000000 0.0517164700000000 0.00116696000000000 0.0557495900000000 0.0914988300000000 -1.13997600000000 0.0414240200000000 0.0577823100000000 0.0105785100000000 0.00108611000000000 -0.0288816700000000;0.0615821900000000 0.0583618300000000 0.0511852600000000 0.0289944100000000 0.0324672100000000 0.0399205500000000 0.0120189900000000 -1.40059100000000 0.107321100000000 -0.0418258200000000 0.00794198000000000 0.0169553600000000;0.0492929500000000 0.0474342600000000 0.0408789900000000 0.0231899600000000 0.0259024200000000 0.0318212600000000 0.00968903000000000 0.0491629300000000 -1.06225300000000 -0.0527534900000000 0.00729352000000000 0.0141123100000000;0.0838631200000000 0.0475778400000000 0.0737786900000000 0.0403001400000000 0.0480159300000000 0.0602565500000000 0.0133273100000000 -0.0583065100000000 -0.0873152600000000 -1.09025000000000 -0.0307301500000000 -0.000891240000000000;0.0797492100000000 0.00977151000000000 0.0746899400000000 0.0392298400000000 0.0498881100000000 0.0638530000000000 0.00929280000000000 0.0737743300000000 0.117205400000000 -0.0110758600000000 -1.12388200000000 -0.0275144900000000;0.000883260000000000 -0.00104474000000000 0.000974480000000000 0.000463960000000000 0.000689950000000000 0.000920180000000000 -6.96000000000000e-06 0.000773320000000000 0.00111838000000000 0.000114490000000000 -0.000194340000000000 -0.174181000000000;]
Chris Turnes
Chris Turnes 2014-8-7
编辑：Chris Turnes 2014-8-7
How are you trying to rebuild the matrix from the eigendecomposition? Note that since the matrix is not symmetric, it is not true that V^{-1} = V^H. Therefore, to reconstruct the matrix, you should instead try
>> Ar = V*D/V; % Ar = V*D*V^{-1}
which is the proper eigendecomposition. When I try this with the matrix you posted, I get:
>> Ar = V*D/V; % Ar = V*D*V^{-1}
>> norm(a - Ar)
ans =
2.4546e-15

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