\[\mathbf{a}_{ij}\]
or as a matrix.\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \]
In general the
\[\mathbf{a}_{ij}\]
are not related so there are 3{sup}3{/sup =9 components.
If the i,j can run from 1 to \[n\]
there are \[n^2\]
components.
For a symmetric system \[\mathbf{a}_{ij} = a_{ji}\]
The matrix above becomes
\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{array} \right) \]
There are
\[\frac{3(3-1)}{2} +3=6\]
components.If the i,j can run from 1 to
\[n\]
there are \[\frac{n(n-1)}{2} +n=\frac{n(n+1)}{2}\]
components.
For a skew symmetric system \[\mathbf{a}_{ij} = -a_{ji}\]
All the diagonal elements must be zero.
The matrix above becomes
\[ \left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{array} \right) \]
There are
\[\frac{3(3-1)}{2}=3\]
components.If the i,j can run from 1 to
\[n\]
there are \[\frac{n(n-1)}{2}\]
components.