An easier method to obtain the sphere, cylinder, and axis from an off-axis dioptric power matrix

Dr. W. F. Long pointed out that calculations of decentration in spherocylindrical lenses, as well as calculations of combinations of obliquely crossed spherocylindrical lenses, are considerably simplified by the use of matrix methods. In the obliquely crossed lens problem, Long used eigenvalue techniques to obtain the sphere, cylinder, and axis of the equivalent lens. This paper presents an alternative to the eigenvalue method. This alternative method uses the invariance of the trace and determinant of the dioptric power matrix. This alternative is conceptually easier to understand than the eigenvalue method and perhaps will encourage more people to use the matrix methods.