I know that a matrix stands for some kind of linear transformation. such as $ \left( \begin{matrix} 1&m\\ 0&1 \end{matrix} \right) $ as a shear mapping matrix. There are all kinds of transformations including rotation, reflection, scaling, shear mapping, squeeze mapping and projection.(Are there any more? Please list them out if you can.)
I try to apply some imagination to symmetric matrices, and I need more geometrical or visualizable interpretation, for this specific kind of matrix has so many useful properties.
But as for such a big category of matrices (symmetric matrices), I can't figure out a common interpretation or imagination. For example, $ \left( \begin{matrix} \frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}\\ \end{matrix} \right) $ is a symmetric matrix, and it's a projection matrix. $ \left( \begin{matrix} \frac{1}{2}&0\\ 0&\frac{1}{2}\\ \end{matrix} \right) $ is also a symmetric matrix, but it's a scaling one.
May be there are some more common and stronger interpretation(imagination/representation, anyway) for symmetric matrices, I don't know. May be you have some idea?