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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.

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  • $\begingroup$ This is not a duplicate question. I am not interested in principal minors. I am interested in kth order minors, that is, removing more than 1 row/column. $\endgroup$ – as2457 Mar 29 '18 at 10:53
  • $\begingroup$ Have you read the answer in the duplicate? $\endgroup$ – user1551 Mar 29 '18 at 10:59
  • $\begingroup$ OK, I think I understand. You're saying that what I'm looking for is exactly, $s_{n-k}$, the sum of products of eigenvalues? I will certainly look up this proof. $\endgroup$ – as2457 Mar 29 '18 at 11:10
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    $\begingroup$ Yes. The proof is simple. E.g. see the discussion in Horn and Johnson after definition 1.2.9. $\endgroup$ – user1551 Mar 29 '18 at 11:13
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    $\begingroup$ A lot of people would still call these principal minors, even if they're not $(n-1) \times (n - 1)$ minors. In case you didn't know, the sum $\sum_I \det A_{I,I}$ (where the sum is over subsets containing $n-k$ elements), is equal up to sign to the coefficient of $x^k$ in the characteristic polynomial of $A$. $\endgroup$ – Joppy Mar 29 '18 at 13:00
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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}

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