How to find the matrix exponential in Mathematica?

Question

I'm trying to take the matrix exponential of a skew symmetric rotation matrix, S. I expect that the result is Rodrigues' rotation formula I + sin(theta)*S + (1-cos(theta))*S*S. However, Mathematica returns something that doesn't look like that formula and it's result is trying to take the square root of a negative number.

Here's my code:

S = { { 0, -omegaz, omegay }, {omegaz, 0, -omegax}, {-omegay, omegax, 0} };
FullSimplify[MatrixExp[S]]

That results in Mathematica:

Am I doing something wrong?

Answer 1

The result are the same.

In Rodrigues' rotation formula, the skew matrix is made from the unit vector, therefore you have assumption:

1 == omegax^2 + omegay^2 + omegaz^2

And you need to use:

MatrixExp[theta S]

And if you run:

rod = IdentityMatrix[3] + Sin[theta] S + (1 - Cos[theta]) MatrixPower[S, 2]
rod = FullSimplify[rod, Assumptions -> {omegax^2 + omegay^2 + omegaz^2 == 1}]
expS = FullSimplify[MatrixExp[theta S], Assumptions -> {omegax^2 + omegay^2 + omegaz^2 == 1}]
rod == b
(* True *)

Thus, the Mathematica calculates the rotation matrix correctly.