A concise Putnam linear algebra problem This is a Putnam practice problem a friend gives me:
Show that diagonal $(A_{11}, A_{22},...,A_{nn})$ of a symmetric matrix $A_{n \times n}$ lies in convex hull defined by all $n!$ permutation of $(\lambda_1,\lambda_2,...,\lambda_n) \in \mathbb{R}^n $ where $\lambda_i $ are eigenvalues of $A$.
I am literally without a clue. I think this has to do with sum of diagonal being sum of eigenvalues but not sure how to proceed from here.

Let $\Lambda$ be a diagonal matrix with diagonal entries equal to the eigenvalues $\lambda_1,\dots,\lambda_n$ of $A$, and let $U$ be an orthogonal matrix such that $A = U\Lambda U^T$. We find that
$$
A_{ii} = \sum_{j=1}^n u_{ij}^2\cdot \lambda_j .
$$
There are several ways to go from there, but the standard way to proceed is to apply the Birkhoff-von Neumann theorem or something like it.
