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Identities for other coefficients of the characteristic polynomial

Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots \lambda_n$ and ${\textrm tr}(A) = \lambda_1 + \dots + \lambda_n$. Now let $i_k(A) = e_k(\lambda_1, \dots, \lambda_n)$ (where $e_k$ is the $k$th elementary symmetric function), so that det=$i_n$ and tr=$i_1$. Now I'm wondering:

Is there anything interesting to say about the $i_k$ other than $i_1$ and $i_n$?

For example, for $i_n$, we have the identity $\det(AB) = \det(A) \det(B)$, which is highly non-trivial from a purely algebraic viewpoint. Are there corresponding identities for the $i_k$? I guess this could be formalized as: "are there polynomials $p,q$ such that $i_k(p(A,B)) = q(i_k(A), i_k(B))$ for $k$ different from $1$ and $n$?", although I'm really more interested in general references regarding the $i_k$. Googling "symmetric function eigenvalues" gives no helpful pointers, and I haven't seen the $i_k$ mentioned extensively in any textbook I've read. Of course, $i_k(A) = tr B$ where $B$ is the endomorphism on $\bigwedge^k \mathbb{R}^n$ mapping $u_1\wedge \dots \wedge u_k$ to $Au_1\wedge \dots\wedge Au_n$, and $i_k(A)$ is the sum of all the principal $k \times k$ minors of $A$.

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    As I mention in the other question, my opinion is that the correct generalization of the multiplicativity of the determinant is the functoriality of the entire exterior power. The additivity of the trace strikes me as a somewhat separate phenomenon (among other things it makes sense in a wider context than exterior powers do). But if that doesn't answer your question I can unclose this one.2012-06-03
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    Thanks for your answer (the old one in particular)!2012-06-03
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    @QiaochuYuan : I would be interested in unclosing this question, since there are generalizations which are not of the kind you mention in your old answer. For example, $i_2(A+B) = i_2(A)+i_2(B) +i_1(A)i_1(B) - i_1(AB)$. This can be found just by comparing coefficients of det(I+tX) for various X, but I have not found a 'unified' answer (expressing the analogous thing for i_k(A+B) in a conceptual way).2012-07-07
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    Would you mind just asking a new question which lists that as an example and linking to this one? I think it would be cleaner.2012-07-07
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    Fair enough. http://math.stackexchange.com/questions/167978/2012-07-07

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