## Number of Independent Elements in a Symmetric Matrix

The entries in a symmetric matrix satisfy
$a_{ij} = a_{ji}$
.
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.