The concepts of matrix plane, step characteristic, zero coincidence point, etc., are defined and the main properties of the matrix plane are emphasized. A criterion permitting one to prove that no step characteristics exist in a given region of the matrix plane is stated. Thus, using some configuration changes of the step characteristics in the matrix plane, a lot of steady-state synthesis problems (substitution of the unstable limit cycle by a stable one, etc.) may be solved in a simple way. By expressing the matrix-plane components in terms of Chebyshev polynomials, a quasi-optimization procedure is suggested. The procedure permits one to determine the non-linearity vector (and consequently the non-linearity shape), which then enables a desired transient response to be obtained. It is shown that the restrictive conditions in the matrix plane, resulting from such a synthesis procedure, may be associated with a quadratic programming method which is used to compute optimized correction of non-linearities. Thus the matrix-plane concept leads to a general, computer-oriented approach which allows one to solve steady-state as well as transient behaviour problems for large classes of non-linear systems, including the noise perturbed ones.