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What are some elegant proofs to demonstrate that the principal minors of a valid, but singular correlation matrix are all non-negative?

A valid correlation matrix is a square symmetric matrix, has ones on its main diagonal and is positive semi-definite. A singular correlation matrix has a matrix determinant of zero, i.e. at least one eigenvalue equal to zero.

The proof that I have: It can be shown that any $n\times n$ singular correlation matrix can be written as the product of $n$ idempotent matrices. It can also be shown that the eigenvalues of an idempotent matrix are always either $0$ or $1$. Therefore, it is not possible that singular correlation matrices have negative eigenvalues. As a consequence, all principal minors of singular correlation matrices have to be non-negative.

Andreas Steiner
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    Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Jun 24 '21 at 08:47
  • "Therefore, it is not possible that singular matrices have negative eigenvalues " is obviously false e.g. $\begin{bmatrix} -1&0\ 0& 0\end{bmatrix}$ – user8675309 Jul 05 '21 at 17:40
  • This is a singular matrix with a negative value, but it is not a valid correlation matrix. I have corrected my question, thank you. – Andreas Steiner Jul 06 '21 at 13:46
  • you have not corrected it. Your "proof" is that a singular correlation matrix is a product of idempotent matrices and hence can't have negative eigenvalues because idempotent matrices have eigenvalues of 0 and 1. This is not a proof. E.g. it implies all singular matrices cannot have negative eigenvalues which is what my original comment addresses. The standard proof is to use quadratic forms. Cauchy Interlacing also works, which I used to prove a stronger result here: https://math.stackexchange.com/questions/4145638/a-is-positive-semidefinite-iff-textdet-b-k-geq-0/ (ref Direction 1) – user8675309 Jul 06 '21 at 19:08
  • Not, it only has implications for singular correlation matrices. A singular correlation matrix has to be square, has to be symmetrical along the main diagonal, has to have ones one the main diagonal, and is required to be positive semidefinite. The example you gave is not a valid correlation matrix. I am interested in valid correlation matrices only. – Andreas Steiner Jul 07 '21 at 07:08

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