  We present a new Monte Carlo algorithm that allows the simultaneous
determination of a few extremal eigenpairs of a very large matrix. It extends
the power method and uses a new sampling method, the sewing method, that does a
large state space sampling as a succession of samplings from a smaller state
space. We illustrate the new algorithm by its determination of the two largest
eigenvalues of the transfer matrix of a square Ising model at the critical
temperature for sizes from $16\times 16$ to $48\times 48$.
