Observations on the statistical iteration of matrices

Monte Carlo calculations which rely on the statistical iteration of some operator can sometimes lead to results in which the variance grows as the iterations proceed. Alternatively the variance can be stable but the result incorrect. The effects are demonstrated by the Monte Carlo iteration of a 2 2 matrix and analyzed in detail for this simple case. In addition an algebraic formulation of a full Monte Carlo calculation with many simultaneous configurations including a method for keeping the number of configurations constant is given. With this formulation it is shown that the naive sampling of the wave function and naive estimate of the eigenvalue based on the growth in the number of configurations will be stable but biased. It is made plausible that for a sufficiently large number (M) of simultaneous configurations the naive method leads to a result which approaches the correct one as M. It is shown that the correct average eigenvalue and eigenvector is a certain weighted average defined so that it avoids the problem of the growing variance and thus becomes more accurate as the chain is extended.