Shear representations of beam transfer matrices

The beam transfer matrix, often called the $\mathrm{ABCD}$ matrix, is one of the essential mathematical instruments in optics. It is a unimodular matrix whose determinant is 1. If all the elements are real with three independent parameters, this matrix is a $2\times 2$ representation of the group Sp(2). It is shown that a real $\mathrm{ABCD}$ matrix can be generated by two shear transformations. It is then noted that, in para-axial lens optics, the lens and translation matrices constitute two shear transformations. It is shown that a system with an arbitrary number of lenses can be reduced to a system consisting of three lenses.