Possible Duplicate:
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 $A$.