Beam transfer matrix (${\left(\genfrac{}{}{0ex}{}{x}{x^{\prime }}\right)}_2=M{\left(\genfrac{}{}{0ex}{}{x}{x^{\prime }}\right)}_1$)

A beam transfer matrix is a linear small-angle approximation to transverse transport between two positions along a beamline:Here $x$ is transverse displacement and $x^{\prime }=dx/ds$ is trajectory slope. A drift of length $d$ and a focusing thin quadrupole magnet have respective matricesSuccessive elements compose by matrix multiplication, with the first element acting on the right. At fixed longitudinal momentum, ideal uncoupled transport governed by Hamiltonian mechanics preserves transverse phase space area and gives $\det M=1$. During acceleration, the slope coordinate is not a fixed multiple of canonical transverse momentum; the corresponding area in $(x,x^{\prime })$ can decrease through adiabatic damping of geometric emittance.

For a real one-cell matrix with $\det M=1$, its eigenvalues obey ${\lambda }^2-(\mathrm{tr}M)\lambda +1=0$. Thus $|\mathrm{tr}M|<2$ gives a conjugate pair of distinct unit-modulus eigenvalues and bounded repeated-cell motion. If $|\mathrm{tr}M|>2$, one real eigenvalue has magnitude greater than one, producing an unstable direction. At equality, a nontrivial Jordan block can produce linear growth, so unit-modulus eigenvalues alone do not establish boundedness. This is a map for beam optics; the transfer matrix used in combinatorics has a different application.