4
$\begingroup$

In a paper I'm reading I've come accross this notation several times now and can't figure out what it's supposed to represent. Sometimes it is applied to just one term or a product of terms like this and sometimes it seems to be applied to a list of arguments like the occurrence in this matrix.

Can anyone help me understand the meaning of this notation?

$\endgroup$
4
  • $\begingroup$ @MishaLavrov This is the paper: uu.diva-portal.org/smash/get/diva2:827236/FULLTEXT01.pdf $\endgroup$ Commented Apr 17, 2017 at 23:36
  • $\begingroup$ Looks like p.10 says "$[\boldsymbol{a}]_\times$ is the skew-symmetric matrix representation of the vector cross product" which is a more concise and less helpful version of the answer you've already received. But given that the paper has a section for symbols where they define things like transpose I'm annoyed that it's not included there. $\endgroup$ Commented Apr 17, 2017 at 23:44
  • $\begingroup$ Oh and Appendix A spells it out, except that due to an unfortunate typesetting failure it defines $[\boldsymbol{a}]_x$ instead. $\endgroup$ Commented Apr 17, 2017 at 23:46
  • $\begingroup$ I usually write it as $ [ {\bf r} \times ]$ to indicage that the vector $\bf r$ is transformed into the matrix for the cross product operator. $\endgroup$ Commented Apr 23, 2017 at 23:18

1 Answer 1

Reset to default
7
$\begingroup$

Let $V=\begin{pmatrix}a\\b\\c\end{pmatrix}$.

$[V]_{\times}$ denotes the antisymmetrical matrix, associated with the linear operation "taking the cross product with $V$", i.e.,

$$[V]_{\times}X:=V \times X=\begin{pmatrix} -cy+bz\\ \ \ \ cx-az\\-bx+ay \end{pmatrix},$$

which means that $$[V]_{\times}=\begin{pmatrix}\ \ \ 0&-c& \ \ \ b\\ \ \ \ c& \ \ \ 0&-a\\-b& \ \ \ a& \ \ \ 0\end{pmatrix}.$$

(this notation is described in (https://en.wikipedia.org/wiki/Cross_product)).

Remarks:

1) Very often $V$ is assumed to be unitary ($a^2+b^2+c^2=1$).

2) Here is a very interesting paper (https://arxiv.org/pdf/1312.0788v1.pdf) where this notation is used with many of its properties grouped in its Appendix A.

3) This notation doesn't look very old ; maybe one should post the question of its origin (robotics ?) on the "history of mathematics" section of SE.

Edit:

4) I have had a glance at the source paper you have indicated after I wrote my answer.

In the block-defined matrix, $\hat R^n_{b,k}f_k^b$ has to be a vector. There are two ways to consider this notation:

  • a) either as a single matrix $\hat R^n_{b,k}$ or $\left(\hat R^n_{b}\right)_k$applied to a vector $f_k^b$, yielding a vector.

  • b) or (much less likely) a linear combination of matrices applied to vectors, with a hidden summation sign according to Einstein notation (https://en.wikipedia.org/wiki/Einstein_notation), $\sum_{b=1}^m \hat R^n_{b,k}f_k^b$ where the summation is on the index that is present once in a lower, once in a upper position, here index $b$.

5) See also the recent document (https://people.eecs.berkeley.edu/~wkahan/MathH110/Cross.pdf) and the older one (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.825.1726&rep=rep1&type=pdf) (1989)

$\endgroup$
2
  • 1
    $\begingroup$ Thanks, this does seem to be what the notation means. However I'm still a little unclear what the computation taking place in the matrix in my second example is. My first thought would be that the vectors are being multiplied and then the anti-symmetrical matrix is taken, but with the comma I'm not so sure. Is it possible this is just a notational error? $\endgroup$ Commented Apr 18, 2017 at 0:16
  • $\begingroup$ Ah yes of course, apologies, I was pretty tired when I wrote that comment and it didn't occur to me that the comma was within the subscript. Of course R is a matrix being applied to the vector f, yielding a vector result from which the anti-symmetric matrix is taken. $\endgroup$ Commented Apr 18, 2017 at 13:51

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.