I am running numerical simulations on large, sparse, Markov matrices $P$ and want to understand the steady state equilibrium of said Markov matrices.
Since these matrices are Markov, all the columns sum to a value of $1$, and by definition they have an eigenvalue of $\lambda_{max} = 1$. Now upon building these matrices in Matlab, I find that they are very ill-conditioned so when I run eigs(P, 1) I am told that the computation to find the eigenvalue numerically either does not converge, or I get a largest eigenvalue that is close to, but not exactly, $1$ such as $.98$.
I have also tried in Matlab to perform null space calculations on the matrix difference null(P-eye(n)), however for large matrices this operation tends to run out of memory.
Is anyone aware of any numerical programs or techniques (particularly using Matlab) that can efficiently compute the null space or eigenvector of a space matrix, with respect to memory?