= Beam transfer matrix
{title2=$\binom{x}{x'}_{\!2}=M\binom{x}{x'}_{\!1}$}
A linear paraxial map between transverse position and trajectory angle at two beamline locations. A drift of length $d$ has $M=\begin{pmatrix}1&d\\0&1\end{pmatrix}$; a focusing thin quadrupole has $M=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}$. Lossless uncoupled maps have determinant one. Repeated-cell bounded oscillatory motion requires $|\operatorname{tr}M|<2$; at equality a Jordan block can still cause secular growth. This beam-optical map is distinct from a combinatorial transfer matrix.
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