A skew-symmetric matrix has the property:

or expressed differently:

It can be formed from a vector $v = \left ( a_1 a_2 a_3 \right )$ as

For real valued skew-symmetric matrices $A\in \mathbb {R}^{n\times n}$ the diagonal values are $0$ and the eigenvalues are pure imaginary or $0$.

$so(3)$

$so(3)$ is the set of all $3\times 3$ skew-symmetric matrices.

Angular Velocity

$[\omega]$ is the matrix representation of an angular velocity $\omega \in {\mathbb {R}}^{3}$ and is an element of $so(3)$.

Cross product

For the special case $n = 3$ the skew-symmetric matrices can be used to express a vector cross product as a matrix multiplication.

The cross product of two vectors $a\in {\mathbb {R}}^{3}$ and $b\in {\mathbb {R}}^{3}$ can be expressed as:

This allows to differentiate formula with a cross product:

Relation to Rotation Matrices

Where $[\omega_c] \in so(3)$ and $[\omega_b] \in so(3)$ are the angular velocities represented in the the reference frame $\{s\}$ and the body frame $\{b\}$, respectively, as skew-symmetric matrices.

Note:

$R$ and $\dot{R}$ individually depend on both $\{s\}$ and $\{b\}$.

But:

• $\dot{R}R^{-1}$ is independent of $\{b\}$
• $R^{-1}\dot{R}$ is independent of $\{s\}$