Editing Skew-symmetric matrix (section)

=== Cross product ===
Three-by-three skew-symmetric matrices can be used to represent cross products as matrix multiplications. Consider two [[Vector (mathematics and physics)|vectors]] <math>\mathbf{a} = \left(a_1, a_2, a_3\right)</math> and <math>\mathbf{b} = \left(b_1, b_2, b_3\right).</math> The [[cross product]] <math>\mathbf{a}\times\mathbf{b}</math> is a [[bilinear map]], which means that by fixing one of the two arguments, for example <math>\mathbf{a}</math>, it induces a [[linear map]] with an associated [[transformation matrix]] <math>[\mathbf{a}]_{\times}</math>, such that

<math display="block">\mathbf{a}\times\mathbf{b} = [\mathbf{a}]_{\times}\mathbf{b},</math>

where <math>[\mathbf{a}]_{\times}</math> is

<math display="block">[\mathbf{a}]_{\times} = \begin{bmatrix}
      \,\,0 &  \!-a_3 & \,\,\,a_2 \\
  \,\,\,a_3 &       0 &    \!-a_1 \\
     \!-a_2 & \,\,a_1 &     \,\,0
\end{bmatrix}.</math>

This can be immediately verified by computing both sides of the previous equation and comparing each corresponding element of the results.

{{See also|Plücker matrix}}
One actually has
<math display="block">[\mathbf{a \times b}]_{\times} =
  [\mathbf{a}]_{\times}[\mathbf{b}]_{\times} - [\mathbf{b}]_{\times}[\mathbf{a}]_{\times};
</math>

i.e., the commutator of skew-symmetric three-by-three matrices can be identified with the cross-product of two vectors.  Since the skew-symmetric three-by-three matrices are the [[Lie algebra]] of the rotation group <math display="inline">SO(3)</math> this elucidates the relation between three-space <math display="inline">\mathbb{R}^3</math>, the cross product and three-dimensional rotations.  More on infinitesimal rotations can be found below.