CODEWITHSUNDEEP
matlablinear-algebraeigenvectoreigenvalue
Spencer
Spencer

Reputation: 245

Eigenvalues in MATLAB

In MATLAB, when I run the command [V,D] = eig(a) for a symmetric matrix, the largest eigenvalue (and its associated vector) is located in last column. However, when I run it with a non-symmetric matrix, the largest eigenvalue is in the first column.

I am trying to calculate eigenvector centrality which requires that I take the compute the eigenvector associated with the largest eigenvalue. So the fact that the largest eigenvalue appears in two separate places it makes it difficult for me to find the solution.

Upvotes: 9

Views: 20570

Answers (4)

Shai
Shai

Reputation: 114796

If you only care for the eigenvector associated with the largest eigenvalue, isn't it better to use eigs?

[V, D] = eigs( a, 1, 'lm' ); %// get first eigenvector with largest eigenvalue magnitude.

Upvotes: 1

gnovice
gnovice

Reputation: 125854

You just have to find the index of the largest eigenvalue in D, which can easily be done using the function DIAG to extract the main diagonal and the function MAX to get the maximum eigenvalue and the index where it occurs:

[V,D] = eig(a);
[maxValue,index] = max(diag(D));  %# The maximum eigenvalue and its index
maxVector = V(:,index);           %# The associated eigenvector in V

NOTE: As woodchips points out, you can have complex eigenvalues for non-symmetric matrices. When operating on a complex input X, the MAX function uses the magnitude of the complex number max(abs(X)). In the case of equal magnitude elements, the phase angle max(angle(X)) is used.

Upvotes: 5

user85109
user85109

Reputation:

Note that non-symmetric matrices tend to have complex eigenvalues.

eig(rand(7))
ans =
       3.2957              
     -0.22966 +    0.58374i
     -0.22966 -    0.58374i
     -0.38576              
      0.49064              
      0.17144 +    0.27968i
      0.17144 -    0.27968i

Also note that eig does not explicitly return sorted eigenvalues (although the underlying algorithm tends to produce them in a nearly sorted order, based on the magnitude of the eigenvalue), but even if you do do a sort, you need to understand how sort works on complex vectors.

sort(rand(5,1) + i*rand(5,1))
ans =
      0.42343 +    0.51539i
    0.0098208 +    0.76145i
      0.20348 +    0.88695i
      0.43595 +    0.83893i
       0.8225 +    0.91264i

Sort, when applied to complex inputs, works on the magnitude of the complex number.

Upvotes: 4

Amro
Amro

Reputation: 124563

What I usually do is:

[V D] = eig(a);
[D order] = sort(diag(D),'descend');  %# sort eigenvalues in descending order
V = V(:,order);

Upvotes: 16

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