The linear parameters and the decoupling matrix for linearly coupled motion in 6 dimensional phase space

It will be shown that starting from a coordinate system where the 6 phase space coordinates are linearly coupled, one can go to a new coordinate system, where the motion is uncoupled, by means of a linear transformation. The original coupled coordinates and the new uncoupled coordinates are related by a 6×6 matrix, R. It will be shown that of the 36 elements of the 6×6 decoupling matrix R, only 12 elements are independent. A set of equations is given from which the 12 elements of R can be computed from the one period transfer matrix. This set of equations also allows the linear parameters, the β_i , α_i, i=1, 3, for the uncoupled coordinates, to be computed from the one period transfer matrix