$[ \cdot ]_{\times}$ operator meaning

Question

In reading articles about stereo image rectification, I've come across the operator $[ \cdot ]_{\times}$ in a few places. One example is here where it says

When we have determined a decomposition $E = SR$ we need to compute a translation vector $t$ from $S$ such that $[t]_{\times} = S$. For such a $t$ we have $St = [t]_{\times} t = t × t = 0$. Therefore the vector $t$ is in the null space of $S$.

I get the impression that it must have something to do with the cross product or orthogonality.

In another paper I see it show up again as:

The fundamental matrix corresponds to two stereo-rectified images if and only if it has the special form (up to a scale factor):

$$[i]_{\times} = \left[ \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right] =\left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{array} \right] $$

(NOTE: The fundamental matrix is a 3x3 orthonormal matrix).

What does $[\cdot]_{\times}$ mean?

Answer 1

I found the answer inadvertently on Wikipedia: https://en.wikipedia.org/wiki/Cross_product#Conversion_to_matrix_multiplication

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector:

$$a \times b = [a]_{\times}b = \left[ \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0\end{array} \right]\left[ \begin{array}{c} b_1 \\ b_2 \\ b_3\end{array} \right] $$

                                              ...

$$[a]_{\times} := \left[ \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0\end{array} \right] $$

Answer 2

It's not standard notation as far as I'm aware, but from your two quotes it looks like $[v]_\times$ is defined to mean the operator $F$ such that $F(w)=v\times w$ for all $w$. (Or perhaps minus that, I can never remember the rule for cross product signs).