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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.) – Martin Thoma Sep 12 '12 at 5:50
up vote 4 down vote accepted

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.

In my experience however, in most applications of the characteristic polynomial other than just for searching eigenvalues, the fact that it is monic is of importance. (One such application is to show that certain values are algebraic integers over the base ring, i.e., solutions of a monic polynomial equation.) For sure, in those applications monic-up-to-a-sign will be easily seen to do the job as well, but it is more convenient if the characteristic polynomial is just monic, period.

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".

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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 "$-\lambda$".

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