  A generalized version of the Kato-Bloch perturbation expansion is presented.
It consists of replacing simple numbers appearing in the perturbative series by
matrices. This leads to the fact that the dependence of the eigenvalues of the
perturbed system on the strength of the perturbation is not necessarily
polynomial. The efficiency of the matrix expansion is illustrated in three
cases: the Mathieu equation, the anharmonic oscillator and weakly coupled
Heisenberg chains. It is shown that the matrix expansion converges for a
suitably chosen subspace and, for weakly coupled Heisenberg chains, it can lead
to an ordered state starting from a disordered single chain. This test is
usually failed by conventional perturbative approaches.

Computational Physics;Strongly Correlated Electrons