Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5 x 5 augmented system matrix

The matrix method of tracing a paraxial ray through a coaxial optical system that contains spherical refracting elements requires a 2 x 2 system matrix. This paper shows that a suitable 5 x 5 system matrix enables one to apply the same method in the general case of systems that may be noncoaxial and that may contain astigmatic elements. The systems may contain prisms and decentered astigmatic lenses, for example.