Cross Product

In space $\mathbb{R}^3$, it is possible to transform the cross-product operator into a linear mapping, that is, to give a matrix representation of the cross product, such that $[\mathbf{x}]_{\times}\mathbf{y} = \mathbf{x} \times \mathbf{y}$.

In this text, $[\mathbf{x}]_{\times}$ denotes the $3 \times 3$ matrix associated with the cross product. The form of this antisymmetric matrix is

$$[\mathbf{x}]_{\times} = \begin{bmatrix} 0 & -x_2 & x_1 \\ x_2 & 0 & - x_0 \\ - x_1 & x_0 & 0 \end{bmatrix}$$ (1.95)

where $\mathbf{x} = (x_0, x_1, x_2)^{\top}$. This matrix has zero determinant and maximum rank 2.