Dynamics of two by two symmetric matrices of trace zero

In this paper, we describe the entire structure of the vector space $Sy{m}_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sy{m}_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sy{m}_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sy{m}_2^0$.