\[a_{ij} = a_{ji}\]
.Any symmetric matrix has reflectional symmetry in the main diagonal.
Any 3 x 3 symmetric matrix takes the form
\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{31} & a_{23} & a_{33} \end{array} \right) \]
.The independent entries are
\[a_{11} , \: a_{12} \: a_{13}, \: a_{22}, \: a_{23} , \: a_{33}\]
.There are 6 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 and the \[n\]
elements on the leading diagonal to be the independent elements, so there are \[n+ \frac{n(n-1)}{2} = \frac{n(n+1)}{2}\]
independent elements altogether.