# Skew-Symmetric Matrix

A skew-symmetric matrix $A$ is a square matrix whose *transpose equals its negative*, i,e.
$A_{T}=−A$

This is used a lot in Robotics. Not to be confused with Orthogonal Matrix, where the transpose is equal to its inverse.

We use the $_{∧}$ operator to write $a$ as a skew-symmetric matrix.

#### Example of use (from Visual SLAM book)

The Cross Product can be written like this:

\mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\
{{a_1}}&{{a_2}}&{{a_3}}\\
{{b_1}}&{{b_2}}&{{b_3}}
\end{array}} \right\| = \left[ \begin{array}{l}
{a_2}{b_3} - {a_3}{b_2}\\
{a_3}{b_1} - {a_1}{b_3}\\
{a_1}{b_2} - {a_2}{b_1}
\end{array} \right] = \left[ {\begin{array}{*{20}{c}}
0&{ - {a_3}}&{{a_2}}\\
{{a_3}}&0&{-{a_1}}\\
{-{a_2}}&{{a_1}}&0
\end{array}} \right] \mathbf{b} \buildrel \Delta \over = { \mathbf{a}^ \wedge } \mathbf{b}
\end{equation}$$
The result of the outer product is a vector whose direction is perpendicular to the two vectors, and the length is $\left | \mathbf{a} \right | \left | \mathbf{b} \right | \sin \langle { \mathbf {a}, \mathbf {b}} \rangle$, which is also the area of the quadrilateral of the two vectors.
$$\begin{equation} \mathbf{a}^\wedge = \left[ {\begin{array}{*{20}{c}} 0&{-{a_3}}&{{a_2}}\\ {{a_3}}&0&{ - {a_1}}\\ { - {a_2}}&{{a_1}}&0 \end{array}} \right] \end{equation}$$