Find the eigenvectors of ill-conditioned matrix

During numerical simulation of continuous time non-homogeneous Markov process, my propagator matrix becomes ill-conditioned, that is, the smallest eigenvalues exponentially decrease and become indistinguishable at numerical precision accuracy. The calculation of eigenvalues and eigenvectors of ill-conditioned matrices becomes very unstable and regular diagonalization techniques fail and produce bad and unstable results.

All I'm interested in is the (single) eigenvector that corresponds to the largest (unique) eigenvalue of the matrix (the largest eigenvalue is 1 and it's unique. Other eigenvalues are non-negative). All the decomposition methods emphasize the eigenvalues over the eigenvectors. I don't have any interest in other eigenvectors or eigenvalues.

By Perron–Frobenius theorem this largest eigenvector has all positive entries, while other eigenvectors have at least one negative entry. The propagator matrix is a square matrix with all real entries, but it's not orthogonal (or normal) matrix.

I'll be glad to get a reference to technique to calculate numerically the eigenvector that corresponds to the largest eigenvalue.

• Whatever your method, the behavior of the small eigenvalues should make no difference as long as there are no eigenvalues that are "numerically close" to the largest eigenvalue. Apr 18, 2021 at 12:58
• @BenGrossmann It is quite well separated, but practically, MATLAB gives nonsense solution for this eigenvector when the matrix is ill conditioned. If the problem was just with the smaller eigenvalues and their eigenvectors I wouldn't care at all. So practically the solutions are somehow coupled. Apr 18, 2021 at 14:53
• I find that very surprising. You're just using the standard eig method? Apr 18, 2021 at 14:55

You know a priori that the eigenvalue you're looking for is associated with the eigenvalue $$1$$. The best approach, then, is to simply compute the nullspace of $$M - I$$ where $$M$$ is your propogator matrix and $$I$$ is the identity matrix.