is the transfer matrix between the original beam position and the quadrupole(s), is the transfer matrix of the quadrupole(s), and is the transfer matrix between the quadrupole(s) and the monitor screen. During the quadrupole scan process, and stay constant, and changes with the field strength of the quadrupole(s).

The final beam when it reaches the monitor screen at distance s from its original position can be described as another beam matrix :

The final beam matrix can be calculated from the original beam matrix by doing

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