Angular velocity tensor

In physics, the angular velocity tensor is defined as a matrix T such that:

${\displaystyle {\boldsymbol {\omega }}(t)\times A(t)\mathbf {r} _{0}=T(t)A(t)\mathbf {r} _{0}={\begin{pmatrix}0&-\omega _{z}(t)&\omega _{y}(t)\\\omega _{z}(t)&0&-\omega _{x}(t)\\-\omega _{y}(t)&\omega _{x}(t)&0\\\end{pmatrix}}A(t)\mathbf {r} _{0}.}$

where A is an orientation matrix. It allows us to express the cross product

${\displaystyle {\boldsymbol {\omega }}(t)\times A(t)\mathbf {r} _{0}}$

as a matrix multiplication. It is, by definition, a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements.

See also

• Angular velocity
• Rigid body dynamics