Number of Independent Elements in an Antisymmetric Matrix

The entries in am antisymmetric matrix satisfy  
\[a_{ij} = -a_{ji}\]
.
This means that any entry on the main diagonal must be zero since it satisfies  
\[a_{ii} = -a_{ii}\]
.
Any 3 x 3 antisymmetric matrix takes the form
\[ \left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{array} \right) \]
.
The independent entries are  
\[ a_{12} \: a_{13}, \: a_{23} \]
.
There are 3 independent elements. We cab take the three elements in the upper right and the three elements on the leading diagonal to be the independent elements.
In general for an  
\[n x n\]
  square matrix we can take the  
\[\frac{n(n-1)}{2}\]
  elements in the upper right to be the independent elements, so there are  
\[ \frac{n(n-1)}{2}\]
  independent elements.

Add comment

Security code
Refresh