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.
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.