Matrix

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 ${}^{\wedge }$ operator to write $a$ as a skew-symmetric matrix.

Example of use (from Visual SLAM book)

The Cross Product can be written like this:

\begin{equation}\mathbf{a} \times \mathbf{b} = \left\| { \begin{array}{*{20}{c}} \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 $\left|\mathbf{a}\right|\left|\mathbf{b}\right|\sin ⟨\mathbf{a},\mathbf{b}⟩$, 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}