Abstract: The dynamics of a discrete system is represented by the state transition matrix. In this work we used a new form for this matrix: the Non-modal Matrix. A comparative analysis to estimate this matrix is carried out in a very simple linear system by four different methods: Euler forward, Euler backward, Crank-Nicolson, and truncated non-modal matrix. The non-modal matrix solution is used to integrate the spatial-discretized DYNAMO model, a meteorological primitive system capable of simulating several phenomena of importance in atmospheric dynamics. The periodic boundary conditions introduce a block circulant structure and the non-linear terms are additioned to the system by a convolution between these ones and the particular matrix solution. The qualitative behaviour of the model dynamics is performed on the discretized operator spectrum. By means of the non-modal approach we can find the stationary invariant that is associated to only one real eigenvalue.