# Does the sign of the characteristic polynomial have any meaning?

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?

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What do you mean by the sign of a polynomial? –  Gerry Myerson Sep 12 '12 at 5:38
+ or -, e.g. $p_A(\lambda) = +(\lambda-1)^4$ or $p_A(\lambda) = -(\lambda-1)^4$ (as every polynomial in $\mathbb{C}[X]$ can be written as a composition of linear factors I can have a - outside of those linear factors ... but now I see the problem with my question. Thanks.) –  moose Sep 12 '12 at 5:50

Actually the characteristic polynomial is often defined as $$\chi_A=\det(I_nX-A)\in k[X]$$ so as to be always monic (of degree $n$); see for instance in Wikipedia. This differs by a sign (and by calling the identity matrix by the more ususal name of $I_n$) from the definition you cited. The fact that the two contradicting definitions coexist shows that the matter of a factor $(-1)^n$ is not considered of great importance.
Also consider the statement "the coefficient of degree $n-i$ of $\chi_A$ is the $i$-th symmetric function of minus the eigenvalues of $A$ (taken with thir algebraic multiplicities)". With the definition you gave, you'd need to throw in another "minus".
If you know the sign of the leading term, you know wether $n$ is even or odd. For all practical purposes, one might just as well have used $\det(\lambda E-A)$ as definition. However, it looks of course simpler to take a matrix and edit only the diagonal entries by appending "$-\lambda$".