# Eigenvalues of the principal submatrix of a Hermitian matrix

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:

Proposition. Let $\lambda_k(\cdot)$ denotes the $k$-th smallest eigenvalue of a Hermitian matrix. Then $$\lambda_k(A)\le\lambda_k(B)\le\lambda_{k+n-r},\quad 1\le k\le r.$$
• They gave the Matrix Analysis book the volume number $\exp(-1)\approx0.3679$? How cute! – Marc van Leeuwen Feb 24 '16 at 11:01