Diagonalization in eigs with Generalized Eigenvalue Problem with Positive Semidefinitive matrix

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Thiago Morhy
Thiago Morhy 2020-11-20
评论： Torsten 2025-2-15
After I execute an eigs command in Matlab 2020b, using as input matrix A and B, i.e. a generalized eigenvalue problem, and 'SM' as sigma, it appears that unstable eigenvectors are obtained when A is a positive semidefinitive matrix, eventhougth the output eigenvalues are fine. The function I use is:
[V, D] = eigs(A, B, ArbitraryNumberOfEigenvalues, 'SM');
In short, the mathematical problem I'm coding is to model the response of a Finite Element Method Vibro-Acoustic problem, hence if we normalize the eigenvectors in respect to the B matrix and diagonalize the A matrix, we should be obtaining again the eigenvalues.
%% Normalization
nm = size(D, 1);
for j = 1 : nm
fm = V(:, j).' * B * V(:, j);
V(:, j) = V(:, j) / sqrt(fm);
end
%% Diagonalization
D = V.' * A * V;
But, as I said, the curve becomes really unstable:
Now, if I impose a boundary condition to the matrices, as example, excluding the rows and collums from 49th to 72th, A matrix becomes Positive Definite and the curve congerve smoothly:
I believe both curves should converge smoothly. Unfortunalety, I can't just use the output eigenvalues matrix, because I will use the eigenvectors to multiply with other matrices. Is this instability expected ? Is there any workaround ?
Thanks.
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Bohan
Bohan 2025-2-15
What is the output created by eig or eigs in Matlab if the B matrix is not strictly positive definite? I tried some examples and there are output but I am not sure what this means.

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Thiago Morhy
Thiago Morhy 2020-11-22
Cristine Tobler's comment answered my problem.
Yes, eigs was giving me an alert of singularity of matrix A. Using, now, the command:
sigma = -100;
[V, D] = eigs(A, B, ArbitraryNumberOfEigenvalues, sigma);
I obtain the following curve of eigenvalues:
It converges flawlessly. And as the lower eigenvalue is approximately zero, a -1 sigma works as well. Thanks for the help.

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