The classical 2x2 matrix approach to geometrical optics is very limited for practical use, because fundamental decompositions of system matrices do not yield matrices with physically significant properties. The use of permuted matrices proposed by us previously (J. Opt. Soc. Am. 73, 1350-1359) yields a much more powerful representation that is more useful for teaching from beginning undergraduate to advanced graduate levels. The matrices of the previous theory are still valid, but when they are treated in what might be called the focal plane representation, the matrices obtained by the LDU decomposition have a simple and direct physical meaning. The relationship between the older matrix theory and this one is analogous to the relationship between the Descartes and the Newton formalisms of geometrical optics: matrix components are simplified by measuring all distances from the foci. This facilitates synthesis problems, for which the standard approach is not well adapted. In addition to simple applications like lenses, mirrors and diopters, the theory can be applied to more complex cases like lenslike media, resonators, Fourier transform systems and phase-conjugate mirrors. This theory can be directly generalized for nonsymmetrical systems using a 4x4 matrix formalism. The other theory, where distances are measured from the principal planes, cannot be generalized for nonsymmetrical systems having no principal planes.