The second orthogonality conditions in the theory of proper and improper rotations.IV. Solution of the trace and secular equations

Abstract

The equation which connects the trace of a rotation matrix and that of its square, and the secular equation for a rotation matrix , both of which are direct res ults of the second orthogonality conditions, are solved by purely analytic methods based on the group property and the periodicity property of rotation matrices. The point is thus made that the well known formulas for the trace of a rotation matrix in terms of the angle of rotation and for the rotation matrix itself in terms of the axis and angle of rotation, are closely related to the algebraic properties of rotation matrices.