Background
Let $\mathbf{A}$ be a general $n\times n$ matrix with real entries. A construction of the usual adjugate $\text{Adj}[\mathbf{A}]$ may be given by \begin{equation} [\text{Adj}[\mathbf{A}]]_{ji} = (-1)^{i+j}\text{Det}[\mathbf{A}_{ij}]. \end{equation} where the $(n-1)\times(n-1)$ submatrix $\mathbf{A}_{ij}$ is obtained from deleting the $i$-th row and $j$-th column. Alternatively, this could be written as a single product of submatrix eigenvalues: \begin{equation} [\text{Adj}[\mathbf{A}]]_{ji} = (-1)^{i+j+n}\prod_{\lambda \in \sigma(\mathbf{A}_{ij})}\lambda = (-1)^{i+j+n}e_{n}(\lambda^{ij}_1,\ldots,\lambda^{ij}_n), \end{equation} where $\sigma(\mathbf{A}_{ij})$ is the eigenspectrum of $\mathbf{A}_{ij}$, $e_{n}$ is the $n$-th elementary symmetric polynomial, and $\lambda_{k}^{ij}$ is the $k$-th eigenvalue of $\mathbf{A}_{ij}$. Conveniently, the relation \begin{equation} \mathbf{A}^{-1} = \frac{1}{\text{Det}[\mathbf{A}]}\text{Adj}[\mathbf{A}] \end{equation} then allows for calculation of this adjugate through the inverse, and using all the tricks and tools developed for the inverse (i.e could be computed in $O(n^3)$ time).
Question
Is there a characterization for the matrix (say for example, a 'higher order' adjugate $\text{Adj}^{k}[\mathbf{A}])$ generated using the sums of products of the eigenvalues of submatrices: \begin{equation} [\text{Adj}^{k}[\mathbf{A}]]_{ji} = (-1)^{i+j+n-k}e_{n-k}(\lambda^{ij}_1,\ldots,\lambda^{ij}_n). \end{equation}
For example, if $n=4$, the usual adjugate element would be given as \begin{equation} [\text{Adj}[\mathbf{A}]]_{ji} = [\text{Adj}^{0}[\mathbf{A}]]_{ij} = (-1)^{i+j+n}\lambda_{1}^{ij}\lambda_{2}^{ij}\lambda_{3}^{ij}, \end{equation} whilst the 'higher order' adjugate elements may be given through \begin{equation} [\text{Adj}^{1}[\mathbf{A}]]_{ji} = [\text{Adj}^{0}[\mathbf{A}]]_{ij} = (-1)^{i+j+n}\big(\lambda_{1}^{ij}\lambda_{2}^{ij} + \lambda_{1}^{ij}\lambda_{3}^{ij} + \lambda_{2}^{ij}\lambda_{3}^{ij}\big), \end{equation} or \begin{equation} [\text{Adj}^{2}[\mathbf{A}]]_{ji} = [\text{Adj}^{0}[\mathbf{A}]]_{ij} = (-1)^{i+j+n}\big(\lambda_{1}^{ij} + \lambda_{2}^{ij} + \lambda_{3}^{ij}\big). \end{equation}
If it can help to simplify matters, $\mathbf{A}$ may be assumed symmetric.
Motivation
If such a characterization (like the inverse for example) existed, it would save a significant amount of computation compared to calculating eigenvalues for every submatrix, which would quickly become cumbersome as $n$ increases - at present obtaining eigenvalues of each (general) submatrices would take $O(n^3)$ time, which needs to be computed for $n^2$ elements, bringing the time up to $O(n^5)$ for a general matrix.
Current possible directions:
I am currently aware that sums of products of the eigenvalues $e_k$ of $\mathbf{A}_{ij}$ may be written as a sum over the principal minors of order $k$ of $\mathbf{A}_{ij}$, but I don't currently see how this could help.
Thanks very much.