Eigenvalues of the principal submatrix of a Hermitian matrix

Question

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:

• Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$
• Relationship of eigenvalues of a diagonal matrix D and $\mathbf{VDV}^{T}$, where V is a semi-orthogonal matrix

Answer 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.