Exponentiating the skew-symmetric matrix

We have the matrix J(\omega) given by

\displaystyle J(\omega) = \begin{bmatrix}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0\end{bmatrix},

where \omega is a unit vector.

Let’s get its powers:

\displaystyle J(\omega)^2 = \begin{bmatrix}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0\end{bmatrix}\begin{bmatrix}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0\end{bmatrix}

\displaystyle J(\omega)^2 = \begin{bmatrix}-\omega_z^2 -\omega_y^2 & \omega_y \omega_x & \omega_z \omega_x \\ \omega_y \omega_x & -\omega_z^2 -\omega_x^2 & \omega_z \omega_y \\ \omega_z \omega_x & \omega_z \omega_y & -\omega_y^2 -\omega_z^2\end{bmatrix}

\displaystyle J(\omega)^3 = \begin{bmatrix}-\omega_z^2 -\omega_y^2 & \omega_y \omega_x & \omega_z \omega_x \\ \omega_y \omega_x & -\omega_z^2 -\omega_x^2 & \omega_z \omega_y \\ \omega_z \omega_x & \omega_z \omega_y & -\omega_y^2 -\omega_z^2\end{bmatrix}\begin{bmatrix}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0\end{bmatrix}

\displaystyle J(\omega)^3 = \begin{bmatrix}0 & -\omega_z(-\omega_z^2 -\omega_y^2) + \omega_z \omega_x^2 & \omega_y(-\omega_z^2 -\omega_y^2) -\omega_y\omega_x^2 \\ \omega_z(-\omega_z^2 -\omega_y^2) - \omega_z \omega_x^2 & 0 & \omega_x \omega_y^2 -\omega_x(-\omega_z^2 -\omega_x^2) \\ -\omega_y(-\omega_z^2 -\omega_y^2) +\omega_y\omega_x^2 & -\omega_x \omega_y^2 +\omega_x(-\omega_z^2 -\omega_x^2) & 0\end{bmatrix}

\displaystyle J(\omega)^3 = \begin{bmatrix}0 & \omega_z(\omega_z^2 +\omega_y^2+\omega_x^2) & -\omega_y(\omega_z^2 +\omega_y^2+\omega_x^2) \\ -\omega_z(\omega_z^2 +\omega_y^2 +\omega_x^2) & 0 & \omega_x(\omega_z^2 +\omega_x^2+\omega_y^2) \\ \omega_y(\omega_z^2 +\omega_y^2+\omega_x^2) & -\omega_x(\omega_z^2 +\omega_x^2 + \omega_y^2) & 0\end{bmatrix}

\displaystyle J(\omega)^3 = \begin{bmatrix}0 & \omega_z & -\omega_y \\ -\omega_z & 0 & \omega_x \\ \omega_y & -\omega_x & 0\end{bmatrix}

\displaystyle J(\omega)^3 = -J(\omega)

Then we have:

\displaystyle J(\omega)^4 = -J(\omega)^2

\displaystyle J(\omega)^5 = J(\omega)

\displaystyle J(\omega)^6 = J(\omega)^2

and so on.

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