Angular velocity tensor

From Wikipedia, the free encyclopedia

It is defined as this matrix,

$\boldsymbol\omega(t) \times A(t)\mathbf{r}_0 = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0 & -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix} A(t)\mathbf{r}_0.$

It has the important property that when the cross product

$\boldsymbol\omega(t) \times A(t)\mathbf{r}_0$

is written with the matrix multiplication (A is a orientation matrix), this matrix is a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements,