A New Way to Compute the Rodrigues Coefficients of Functions of the Lie Groups of Matrices

In Theorem 1 we present, in the case when the eigenvalues of the matrix are pairwise distinct, a direct way to determine the general Rodrigues coefficients of a matrix function for the general linear group $$\mathbf{GL}(n, \mathbb{R})$$ by reducing the Rodrigues problem to the system (7). Then, Theorem 2 gives the explicit formulas in terms of the fundamental symmetric polynomials of the eigenvalues of the matrix. Our formulas permit to consider also the degenerated cases (i.e., the situations when there are multiplicities of the eigenvalues) and to obtain nice determinant formulas. In the cases n = 2, 3, 4, the computations are effectively given, and the formulas are presented in closed form. The method is illustrated for the exponential map and the Cayley transform of the special orthogonal group SO(n), when n = 2, 3, 4.