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

A linear paraxial map between transverse position and trajectory angle at two beamline locations. A drift of length $d$ has $M=\left(\begin{array}{cc}1& d\\ 0& 1\end{array}\right)$; a focusing thin quadrupole has $M=\left(\begin{array}{cc}1& 0\\ -1/f& 1\end{array}\right)$. Lossless uncoupled maps have determinant one. Repeated-cell bounded oscillatory motion requires $|\mathrm{tr}M|<2$; at equality a Jordan block can still cause secular growth. This beam-optical map is distinct from a combinatorial transfer matrix.