What does $[\mathbf{a}]_\times\mathbf{b}$ mean?

Question

The left means $A$ cross $B$. What does the right mean?

$$\mathbf{a}\times\mathbf{b}=[\mathbf{a}]_\times\mathbf{b}$$

It came from the GitHub post "How to derive essential matrix from camera projection matrices?" by Chen Feng (simbaforrest).

Answer 1

The above transforms a vector function into a linear algebra matrix product

$$ \begin{pmatrix} a_x \\ a_y \\ a_z \end{pmatrix} \times \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix} = \begin{bmatrix} 0 & -a_z & a_y \\ a_z & 0 & -a_x \\ -a_y & a_x & 0 \end{bmatrix} \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix} $$

My personal preference is to include the cross operator inside the brackets, such as

$$ \boldsymbol{a} \times \boldsymbol{b} = [ \boldsymbol{a} \times] \boldsymbol{b} $$

because the $[\boldsymbol{a}\times]$ stands on its own and it reminds me of a matrix.

Alternative notation I have seen is the tilde on top, such as $ \boldsymbol{\tilde{a}} \boldsymbol{b}$, but again this is not clear enough IMHO.

See also Wikipedia for the conversion of the vector cross product to matrix multiplication.