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This question aims at creating an "abstract duplicate" of various questions that can be reduced to the following:

Let $A$ be an $n\times n$ Hermitian matrix and $B$ be an $r\times r$ principal submatrix of $A$. How are the eigenvalues of $A$ and $B$ related?

Here are some questions on this site that can be viewed as duplicates of this question:

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Proposition. Let $\lambda_k(\cdot)$ denotes the $k$-th smallest eigenvalue of a Hermitian matrix. Then $$ \lambda_k(A)\le\lambda_k(B)\le\lambda_{k+n-r},\quad 1\le k\le r. $$

This is a well-known result in linear algebra. Since the usual proof is just a straightforward application of the celebrated Courant-Fischer minimax principle, we shall not repeat it here. See, e.g. theorem 4.3.15 (p.189) of Horn and Johnson, Matrix Analysis, 1/e, Cambridge University Press, 1985.

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    $\begingroup$ They gave the Matrix Analysis book the volume number $\exp(-1)\approx0.3679$? How cute! $\endgroup$ – Marc van Leeuwen Feb 24 '16 at 11:01

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