I am interested in computing the sum of all digonal kth order minors of a matrix.
By this I mean that I have an $n\times n$ matrix $A$, and I define the diagonal kth order minors as determinants of the matrices $A_{I,I}$ where $I$ is some ordered set of $n-k$ non-repeating integers between $1$ and $n$. I then want to compute \begin{equation} \sum_{I} \det A_{I,I}, \end{equation} over all distinct integer sets $I$ with values between $1$ and $n$.
I can of course compute these minor determinants 1 by 1 and sum but I was wondering if this quantity reduces to some property of the full original matrix $A$, or whether it can be computed via matrix multiplication and traces, rather than computing a large number of determinants.
For some context: I am a physicist computing free-fermion correlators which can be computed via determinants. The expression I am trying to compute corresponds to tracing over all intial and final states with a fixed fermion filling.
This answer was provided in the comments of Sum of principal minors as was kindly pointed out above.
The answer is: The sum of all $k^\text{th}$ principal minors of a matrix $A$ is equal to the $k^\text{th}$ elementary symmetric polynomial of its eigenvalues.
More explicitly, a $k^\text{th}$ principal minor is the determinant of a $k\times k$ submatrix of $A$ formed by removing the same rows as columns. The elementary symmetric polynomials $e_k(\{\lambda_i\})$ are the sums of products of all disctint products of $\{\lambda_i\}$ of length $k$. As an example \begin{equation} e_2(\lambda_1, \lambda_2, \lambda_3,\lambda_4) = \lambda_1 \lambda_2 + \lambda_1 \lambda_3 + \lambda_1 \lambda_4 + \lambda_2 \lambda_3 + \lambda_2 \lambda_4 + \lambda_3 \lambda_4. \end{equation}