# Matrix operations preserving the middle coefficients of characteristic polynomial

Let $$n$$ be a positive integer. For a ring $$A$$ and matrix $$M \in \mathrm{Mat}_{n \times n}(A)$$, let $$\chi_{M}(t) = \det(M-t \operatorname{id}_{n}) = (-1)^{n}(t^{n} - \sigma_{M,1}t^{n-1} + \dotsb + (-1)^{n}\sigma_{M,n}) \in A[t]$$ be the characteristic polynomial. Here $$\sigma_{M,1}$$ is the trace of $$M$$ and $$\sigma_{M,n}$$ is the determinant of $$M$$. Recall that $$\sigma_{M,1}$$ is additive and $$\sigma_{M,n}$$ is multiplicative, i.e. $$\sigma_{M_{1}+M_{2},1} = \sigma_{M_{1},1} + \sigma_{M_{2},1}$$ and $$\sigma_{M_{1}M_{2},n} = \sigma_{M_{1},n}\sigma_{M_{2},n}$$.

Are there any matrix operations that preserve the middle coefficients $$\sigma_{M,2},\dotsc,\sigma_{M,n-1}$$ in any reasonable sense?

This is related to the question Geometric interpretation of characteristic polynomial but I don't yet understand whether it answers my question.

• Given that these coefficients are the traces of the exterior powers of $M$, their behavior should be related to identities in $\lambda$-rings. Jan 19, 2019 at 5:49