Eigenvalues of the principal submatrix of a Hermitian matrix

linear-algebra faq matrices eigenvalues-eigenvectors

Solution 1:

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

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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