The antisymmetric indexing function can be used to construct tables and rtable objects of type Array or Matrix.

The antisymmetric indexing function is most commonly used as a parameter to the Matrix constructor when creating antisymmetric Matrices (i.e., matrices where the (i, j)th element is the negative of the (j, i)th element).

The name skewsymmetric is equivalent to antisymmetric in the construction of Matrices (rtable-based).

In general, this indexing function specifies that all entries of an object whose indices are even permutations of each other are the same. All entries whose indices are odd permutations of each other are additive reciprocals. Entries with repeated indices are 0.

The indices are rearranged according to a system-determined canonical ordering.

The syntax for specifying the antisymmetric indexing function varies for tables and rtable objects. For details, see the following examples.

$M≔\mathrm{Matrix}\left(3\,3\,\mathrm{shape}=\mathrm{antisymmetric}\right)$

$M≔\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$M\left[1,2\right]≔5$

$M_{1,2}≔5$

$M$

$\left[\begin{array}{ccc}0& 5& 0\\ \mathrm{-5}& 0& 0\\ 0& 0& 0\end{array}\right]$

$A≔\mathrm{Array}\left(\mathrm{antisymmetric}\,1..3\,1..3\right)$

$A≔\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

$A\left[1,2\right]≔3$

$A_{1,2}≔3$

$A$

$\left[\begin{array}{ccc}0& 3& 0\\ \mathrm{-3}& 0& 0\\ 0& 0& 0\end{array}\right]$

$B≔\mathrm{table}\left(\mathrm{antisymmetric}\right)$

$B≔table\left(\mathrm{antisymmetric}\,\left[\right]\right)$

$B\left[i,j\right]$

$B_{i,j}$

$B\left[j,i\right]$

$-B_{i,j}$

$B\left[i,j,k\right]+B\left[i,k,j\right]$

$0$