I know that the trace of an $n \times n$ matrix is invariant and results from adding the diagonal entries. I also know that the characteristic polynomial of that very same matrix is also invariant, and somehow that invariance seems more primitive than that of the trace.
What is the relation between those two invariances?
Does the trace derive from a primordial invariant represented by the underlying polynomial?
How exactly can that relation be characterized, both logically and numerically? I heard sometime that the trace merely multiplies the polynomial by some factor ($\times (n-1)$) or something along those lines, I really do not recall.
Could you please elaborate on that? Thanks in advance.
Assuming that you define the characteristic polynomial $P(\lambda)$ of the matrix $A$ as $P(\lambda)=\det(A-\lambda\operatorname{Id}_n)$, then $\operatorname{tr}A$ is the coefficient of $\lambda^{n-1}$ in $P(\lambda)$ times $(-1)^{n-1}$. Therefore, the invariance of $P(\lambda)$ implies the invariance of $\operatorname{tr}A$.