# Eigenvalues of the principal submatrix of a Hermitian matrix

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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?

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## 1 Answer

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.

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