Expressing the trace of $A^2$ by trace of $A$ Let $A$ be a a square matrix. Is it possible 
to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least
something close?

It is not. Consider two $2\times 2$ diagonal matrices, one with diagonal $\{1,-1\}$ and one with diagonal $\{0,0\}$. They have the same trace, but their squares have different traces. 
