New unique and concise translation matrices are derived for evaluating the aberration variations of conceptual and real lenses when the paraxial marginal and chief ray paths are arbitrarily changed. They are helpful in investigating the general behaviors of lenses and optimizing the balanced aberrations of lens components in prime and zoom lenses. These new matrices, with a dimension of for monochromatic aberrations and a dimension of for chromatic aberrations, are derived based on our earlier algorithms of which four cases with distinct translation factors and matrices are required according to the relationships of the original and new positions of the object and pupil; otherwise, division-by-zero errors or insufficient numerical accuracy will be encountered. As a comparison, the new matrices have several advantages. First, by introducing four meaningful equivalent optical invariants, multiplying the old matrices, and simplifying the new matrices, they have concise expressions to significantly reduce the calculation time. Second, they are unique and always accurate to apply for all kinds of object and pupil positions without suffering any mathematical problems, i.e., four separated algorithms are no longer necessary. Third, due to the unique property, the component contributions of original aberrations to new aberrations can be directly evaluated and analyzed.
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