Using non-positive (semi)definite matrix as "covariance matrix" for MALA

If the target distribution is a Gaussian distribution, MALA (metropolis-adjusted langevin algorithm) becomes:

$$X_{t+1} = A X_t + \sqrt{2\tau}\xi$$ where $$A= I - \tau\Sigma^{-1}$$.

where $$\tau$$ is a step size, $$\Sigma$$ is the covariance matrix of the target Gaussian distribution, and $$\xi$$ is an i.i.d. standard normal distribution.

I am wondering the following:

If we arbitrarily generate a matrix $$A$$ with the only constraint being convergent (i.e. can be non-positive definite), and run the dynamics shown in the first equation what would be the stationary distribution of $$X_t$$ would become at large $$t$$? More precisely, what would be the covariance matrix of the stationary distribution? Obviously, we cannot simply rearrange $$A= I - \tau\Sigma^{-1}$$ and compute $$\Sigma = \tau (I-A)^{-1}$$, since $$\Sigma$$ wouldn't be positive definite in that case.