Some problems and confusions related to matrix analysis for Gaussian beam reflection are clarified. We can choose right-handed coordinates before and after mirror reflection which is called the traditional coordinate system in this Letter. We can also choose a coordinate system with chirality inversion which is called a novel coordinate system in this Letter. The matrix describing optical components should coincide with the selected coordinate system; errors or confusions will appear otherwise. Spherical mirror reflection and coordinate rotation in nonplanar ring resonators are used to clarify some confusion in previous publications due to disorder of the coordinate system. This work is significant to both Gaussian beam propagation analysis and ring laser resonator design.