Trace of square matrices There are posts in mathoverflow and math.stackexchange about geometric interpretation and intuition of the trace of a matrix . But unfortunately they are not comprehensive and I haven't made any progress in understanding what really is the trace and how it can help us . By the way what is the rationale behind the definition of trace ? what is this mathematical object ?what are it's interpretations?
 A: This might be helpful. If not, it is at least interesting and related.
The Trace-Determinant Plane: Let  $$A=\begin{pmatrix}
        a & b \\
        c & d \\
        \end{pmatrix}.$$
The eigenvalues are the roots of the following characteristic polynomial $$\lambda^2-(a+d)\lambda+(ad-bc)=0.$$
We know that $\det(A)=ad-bc$ which is the constant term in the characteristic polynomial and $\mathrm{tr}(A)=a+d$ which is the coefficient of $\lambda$ in the characteristic polynomial. Thus we can rewrite the characteristic polynomial as $$\lambda^2-\mathrm{tr}(A)\lambda+\det(A)=0.$$ Using the quadratic equation we can solve for $\lambda$ and obtain 
$$\lambda_±={1\over 2} \left(\mathrm{tr}(A)±\sqrt{\mathrm{tr}(A)^2-4\det(A)}\right).$$
The sum of the eignevalues $\lambda_++\lambda_-=\mathrm{tr}(A)$ and the product of the eigenvalues $\lambda_+\lambda_-=\det(A)$. Letting $T=\mathrm{tr}(A)$ and $D=\det(A)$. If we know the values of $T$ and $D$ we can study the geometry of systems like $X'=AX$. That is, we can study the phase portraits of these systems. If $T\lt0$ we have a spiral sink, if $T\gt0$ we have a spiral source, and if $T=0$ we have a center.
This gives a complete overview of all of the possible behavoirs of $2 \times 2$ linear systems. 
