Matching of eigenvalues of 2 matrices

Hello Everybody,

Suppose I have 2 matrices with same size: . is a slight change of in the form of: . I do eigenvalue analysis on both matrices:

lambda_1 = eig(A_1);
lambda_2 = eig(A_2);

and want to do a comparison between eigenvalues of each matrix. How can I find the matching eigenvalue for the original , i.e. how can I identify each eigenmode from to its equivalent in ?

Answer 1

[Christine Tobler]

With R2018a, there is a new function matchpairs which might be useful for this. Basically, it takes a matrix of similarities between two sets, and matches these sets up in pairs.

Here's an example:

>> X = randn(100);
>> d = eig(X);
>> d2 = eig(X + randn(100)*1e-2);
>> max(abs(d - d2))
ans =
   17.9169
>> m = matchpairs(abs(d - d2.'), 1e-1); % The columns of m map the elements of d to those of d2
>> max(abs(d(m(:, 1)) - d2(m(:, 2))))
ans =
    0.1569

The second input to matchpairs gives a cutoff: If two values have a difference larger than 1e-1, they would not be matched at all, and would show up as a new eigenvalue and a disappeared eigenvalues instead.

[Omar Kamel]

Thanks a lot Christine for your elaborate answer.

Answer 2

[KSSV]

A1 = rand(2) ; 
[v1,d1] = eig(A1) ; 

Columns of v1 gives eigen vectors.........diagonal (diag(d1)) gives you eignvalues. To compare them use isequal. Or you may substract them and get the difference.

[Omar Kamel]

Thanks for your reply!

I have already tried such solutions, but the problem that they are not sorted in the same order (specially with complex eigenvalues), so this direct difference based comparison is not the best method to compare.

[KSSV]

direct difference based comparison is not the best method to compare. Then what else method you have?

[Omar Kamel]

what is meant by direct difference based comparison is that not always the 4th eigenvalue from matrix appears also 4th in eigenvalues of matrix but it might appear 7th as an example and the 7th appears 4th, so they might be the same eigenmodes but appearing in different locations. So subtracting the vector of eigenvalues from each other doesn't give what I expect because the eigenvalues are not sorted as each other.

[KSSV]

So sort them and keep them in order.....

[Omar Kamel]

It is not about sorting the list. It is about matching of the eigenvalues and eigenvectors to their equivalent in the other system, to know if any eigenmodes appeared or another eigenmodes disappeared.