Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

The authors show that there is a generalization of Rodrigues’ formula for computing the exponential map exp: so(n)→SO(n) from skewsymmetric matrices to orthogonal matrices when n ≥ 4, and give a method for computing some determination of the (multivalued) function log: SO(n) → so(n). The key idea is the decomposition of a skew-symmetric n×n matrix B in terms of (unique) skew-symmetric matrices B1, . . . , Bp obtained from the diagonalization of B and satisfying some simple algebraic identities. A subproblem arising in computing logR, where R∈SO(n), is the problem of finding a skewsymmetric matrix B, given the matrix B2, and knowing that B2 has eigenvalues −1 and 0. The authors also consider the exponential map exp: se(n)→SE(n), where se(n) is the Lie algebra of the Lie group SE(n) of (affine) rigid motions. The authors show that there is a Rodrigues-like formula for computing this exponential map, and give a method for computing some determination of the (multivalued) function log: SE(n) → se(n). This yields a direct proof of the surjectivity of exp: se(n)→SE(n).