Let $A$ be an $n \times n$ matrix over any base ring, and let the characteristic polynomial of $A$ be given by $x^n + \sum_{i = 1}^n f_ix^{n-i}$. For any $j \in \{1, \dots, n\}$, let $f^{(j)}(x) = x^j + \sum_{i = 1}^{j-1} f_ix^{j-i}$ denote the $j^{\mathrm{th}}$ ``partial characteristic polynomial'' of $A$.
Question: Do the partial characteristic polynomials of $A$ admit a natural algebraic interpretation, or do they arise in a natural way from the matrix $A$? What can be said about the matrix $f^{(j)}(A)$?
The reason I ask the above question is that these partial characteristic polynomials arise naturally in the study of orders in number fields that are defined by polynomials, but I don't understand these polynomials from an algebraic perspective.
What I know:
- We have that $f^{(n)}(x)$ is the characteristic polynomial of $A$ minus $\det(A)$, and the Cayley-Hamilton theorem implies that $f^{(n)}(A)$ is $\det(A)$ times the $n \times n$ identity matrix.
- Testing for several small values of $n$, it appears that each nondiagonal entry of $f^{(n-1)}(A)$ is given by a certain non-principal $(n-1) \times (n-1)$ minor of $A$, whereas the $i^{\mathrm{th}}$ diagonal entry of $f^{(n-1)}(A)$ is the sum of all the principal $(n-1) \times (n-1)$ minors of $A$ except for the minor that arises by deleting the $(n+1-i)^{\mathrm{th}}$ row and column of $A$.
- Testing for several sample values of $n$ and $j$, it appears that the entries of $f^{(j)}(A)$ are linear combinations of the $j \times j$ minors of $A$.
Can anything more specific be said about the polynomials $f^{(j)}(x)$ and/or the matrices $f^{(j)}(A)$? Have these polynomials been studied, and if so, do they have a name?