  The nonlinear dynamics of a system with periodic structure can be analyzed
using a square matrix. We show that because the special property of the square
matrix constructed for nonlinear dynamics, we can reduce the dimension of the
matrix from the original large number for high order calculation to low
dimension in the first step of the analysis. Then a stable Jordan decomposition
is obtained with much lower dimension. The Jordan decomposition leads to a
transformation to a new variable, which is an accurate action-angle variable,
in good agreement with trajectories and tune obtained from tracking. And more
importantly, the deviation from constancy of the new action-angle variable
provides a measure of the stability of the phase space trajectories and tune
fluctuation. Thus the square matrix theory shows a good potential in
theoretical understanding of a complicated dynamical system to guide the
optimization of dynamical apertures. The method is illustrated by many examples
of comparison between theory and numerical simulation. In particular, we show
that the square matrix method can be used for fast optimization to reduce the
nonlinearity of a system.
