The characteristic polynomial of a matrix $A \in \mathbb{C}^{n \times n}$, $p_A (\lambda) = \det(A-\lambda \cdot E)$ can be used to find the eigenvalues of the linear function $\varphi:\mathbb{C}^n \rightarrow \mathbb{C}^n, \varphi(x) := A \cdot x$, as the eigenvalues are the roots of $p_A(\lambda)$. So, for finding the eigenvalues, the sign of the characteristic polynomial isn't important. At the moment, this is to only application of the characteristic polynomial that I know.
Do other applications of the characteristic polynomial exist, where the sign of it is important?
Can I make any statements about the matrix $A$ when I know the sign of its characteristic polynomial?