It is a serious problem. Is there another way to calculate eigenvectors with Matlab or I miss some things?
A serious problem when calculating eigenvectors of a matrix with the function eig
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Hi all,
I calculated the eigenvalues of A=[1 2;3 4] as [vec,val]=eig(A) and I found:vec=[-0.8246 -0.4160;0.5658 -0.9094] and val=[-0.3723 0;0 5.3723].
With Mathematica and Maple, the eigenvalues of A are the same as Matlab, but vec=[0.45743 1.0000;-1.4574 1.0000] (the same in maple and mathematica) is totaly different than Matlab.
It is a serious problem.
Thank you
Réponses (1)
Sebastian Castro
le 17 Août 2015
The results are actually the same, but (not knowing anything about Maple and Mathematica) the formatting is different. It looks like
- The MATLAB vectors are normalized while the Maple/Mathematica ones aren't -- instead, they have one element with a value of 1.0 and the other one is relative to that.
- The MATLAB vectors are expressed in columns while the Maple/Mathematica are in rows.
To test this out, I did the following:
>> v1_maple = [0.45743 1];
>> v1_maple/norm(v1_maple)
ans =
0.4160 0.9094
>> v2_maple = [-1.4575 1];
>> v2_maple/norm(v2_maple)
ans =
-0.8246 0.5657
Compare those results with the columns of the MATLAB results and there you go! Well, one says 0.5657 and the other says 0.5658, but that might be a display/rounding issue.
- Sebastian
4 commentaires
Greg Obi
le 11 Mar 2020
hey can you expound more on how you can get the results to be unormalized. I'm having the same issue and I'm getting closer to understanding it.
Steven Lord
le 11 Mar 2020
By the definition, if V is an eigenvector of a matrix A with eigenvalue d, then A*V = V*d.
Now let's consider a new vector W = 2*V. Is W an eigenvector of A, and if so what's its eigenvalue?
A*W = A*2*V % Since W = 2*V
A*2*V = 2*A*V % Since scalars commute with matrices
2*A*V = 2*V*d % Since V is an eigenvector of A with eigenvalue d
2*V*d = W*d % Since 2*V = W
Therefore A*W = W*d. That means W is also an eigenvector of A and it has the same eigenvalue d as V does.
You can generalize this argument to any non-zero multiple of the eigenvector.
A = rand(5);
[V, D] = eig(A);
v1 = V(:, 1);
d1 = D(1, 1);
% v1 is an eigenvector of A with eigenvalue d1
shouldBeSmall1 = A*v1-v1*d1
% So is pi*v1
shouldBeSmall2 = A*(pi*v1)-(pi*v1)*d1
% We don't need to restrict ourselves to real number either
% (3+4i)*v1 is an eigevector as well
c1 = (3+4i)*v1;
shouldBeSmall3 = A*c1-c1*d1
In Sebastian's example, the chosen non-zero multiplier was (1/V(2)). You can choose a different multiplier if it's appropriate for the physical application the matrices whose eigenvalues you're computing are modeling.
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