I am currently working on a program that aims to take some steady-state probability distribution (specified by particular free parameters of my choosing) and propagates it to some other steady-state (also specified by the same set of modified parameters). I can do this by calculating a tridiagonal matrix (not symmetric, unfortunately) and finding its eigenvalues/eigenvectors, such that:
$$p^{(2)}=\hat{K}^{(2)}(t,t_0)p^{(1)}=V^{(2)}Y^{(2)}U^{(2)}p^{(1)}$$
where the 2's denote the second probability distribution. I let $\hat{K}^{(2)}(t,t_0)$ denote the propagator of the second probability distribution. V is a matrix of the normalized right eigenvectors, U is a matrix of the normalized left eigenvectors, and Y is a diagonal matrix whose entries are not important to our discussion (it is the identity when t=0). The left and right eigenvectors form an orthonormal set and hence I should get that:
$$V^{(2)}U^{(2)}=I$$
I can obtain $U^{(2)}$ from $V^{(2)}$ simply by taking the inverse of $V^{(2)}$, i.e.
$$\big(V^{(2)}\big)^{-1}=U^{(2)}$$
Here is where the issue comes on. $V^{(2)}$ is very close to being a singular matrix. This would be fine if the matrix were not extremely large (300x300-500x500), as MATLAB's Moore-Penrose pseudo-inverse would work beautifully in obtaining the information I need; however, since the matrix is so large, I am not getting results that are necessarily accurate (for example, if I set the parameters that specify the shape of the later probability distribution to a point past a certain threshold, the propagator does not turn into the identity at t=0, as $V^{(2)}U^{(2)} \neq I$ for large portions of the matrix). I believe this error stems from the Moore-Penrose pseudo-inverse algorithm, as it is unable to build a proper inverse of $V^{(2)}$. I am hence curious if there is a way around my predicament - perhaps a better algorithm for calculating pseudo-inverses and/or inverses for matrices that are extremely close to being singular in general.